Ib Maths Hl Vectors Questions 641i1m
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1.
Find the coordinates of the point where the line given by the parametric equations x = 2λ + 4, y = –λ – 2, z = 3λ + 2, intersects the plane with equation 2x + 3y – z = 2. Working:
Answer: ………………………………………….. (Total 4 marks)
2.
Find the value of a for which the following system of equations does not have a unique solution. 4x – y + 2z = 1 2x + 3y
= –6
x – 2y + az = 7 2 Working:
Answer: ………………………………………….. (Total 4 marks)
1
3.
(a)
Find a vector perpendicular to the two vectors:
!
!
!
OP = i – 3 j + 2 k
! ! ! OQ = –2 i + j – k (b)
If OP and OQ are position vectors for the points P and Q, use your answer to part (a), or otherwise, to find the area of the triangle OPQ.
Working:
Answers: (a) ………………………………………….. (b) …………………………………….......... (Total 4 marks)
2
4.
The coordinates of the points P, Q, R and S are (4, 1, –1), (3, 3, 5), (1, 0, 2c), and (1, 1, 2), respectively. (a)
Find the value of c so that the vectors QR and PR are orthogonal. (7)
For the remainder of the question, use the value of c found in part (a) for the coordinate of the point R. (b)
Evaluate PS × PR . (4)
(c)
Find an equation of the line l which es through the point Q and is parallel to the vector PR . (3)
(d) (e)
5.
Find an equation of the plane π which contains the line l and es through the point S. Find the shortest distance between the point P and the plane π.
(4)
(4) (Total 22 marks)
! ! ! ! The vector n = 2 i – j +3 k is normal to a plane which es through the point (2, 1, 2). (a)
Find an equation for the plane.
(b)
Find a if the point (a, a – 1, a – 2) lies on the plane.
Working:
Answers: (a) .................................................................. (b) .................................................................. (Total 4 marks)
6.
The rectangle box shown in the diagram has dimensions 6 cm × 4 cm × 3 cm.
3
H
G
E
F
3cm
D A
C 6cm
B
4cm
Find, correct to the nearest one-tenth of a degree, the size of the angle AHˆ C . Working:
Answer: ....…………………………………….......... (Total 4 marks)
4
7.
Calculate the shortest distance from the point A(0, 2, 2) to the line
! ! ! ! ! ! ! r = 5 i + 9 j + 6 k + t( i + 2 j + 2 k ) where t is a scalar. Working:
Answer: ....…………………………………….......... (Total 4 marks)
5
8.
Consider the points A(l, 2, 1), B(0, –1, 2), C(1, 0, 2), and D(2, –1, –6). (a)
Find the vectors AB and BC . (2)
(b)
Calculate AB × BC . (3)
(c)
Hence, or otherwise find the area of triangle ABC. (2)
(d)
Find the equation of the plane P containing the points A, B, and C. (3)
(e)
Find a set of parametric equations for the line through the point D and perpendicular to the plane P. (2)
(f)
Find the distance from the point D to the plane P. (3)
(g)
Find a unit vector which is perpendicular to the plane P. (2)
(h)
The point E is a reflection of D in the plane P. Find the coordinates of E. (4) (Total 21 marks)
6
9.
The system of equations represented by the following matrix equation has an infinite number of solutions.
⎛ 2 ⎜ ⎜ 1 ⎜ 2 ⎝
−1 2 1
− 9 ⎞ ⎛ x ⎞ ⎛ 7 ⎞ ⎟ ⎜ ⎟ ⎜ ⎟ 3 ⎟ ⎜ y ⎟ = ⎜ 1 ⎟ − 3 ⎟⎠ ⎜⎝ z ⎟⎠ ⎜⎝ k ⎟⎠
Find the value of k. Working:
Answer: .................................................................. (Total 3 marks)
10.
Find a vector that is normal to the plane containing the lines L1, and L2, whose equations are: L1: r = i + k + λ (2i + j – 2k) L2: r = 3i + 2j + 2k + µ (j + 3k) Working:
Answer: .................................................................. (Total 3 marks)
11.
The plane 6x – 2y + z = 11 contains the line x – 1 =
y +1 z − 3 . Find l. = 2 l
7
Working:
Answer: .................................................................. (Total 3 marks)
12.
Let a be the angle between the vectors a and b, where a = (cos θ)i + (sin θ)j, b = (sin θ)i + (cos θ)j and 0 < θ <
π . 4
Express α in of θ. Working:
Answer: .................................................................. (Total 3 marks)
8
13.
(a)
(b)
Given matrices A, B, C for which AB = C and det A ≠ 0, express B in of A and C.
⎛ 1 ⎜ Let A = ⎜ 2 ⎜ 3 ⎝
2 −1 −3
3 ⎞ ⎟ 2 ⎟ , D = 2 ⎟⎠
(i)
Find the matrix DA;
(ii)
Find B if AB = C.
⎛ − 4 13 ⎜ 7 ⎜ − 2 ⎜ 3 − 9 ⎝
− 7 ⎞ ⎟ − 4 ⎟ and C = 5 ⎟⎠
(2)
⎛ 5 ⎞ ⎜ ⎟ ⎜ 7 ⎟ . ⎜10 ⎟ ⎝ ⎠
(3)
(c)
Find the coordinates of the point of intersection of the planes x + 2y + 3z = 5, 2x – y + 2z = 7 and 3x – 3y + 2z = 10. (2) (Total 7 marks)
14.
(a)
If u = i +2j + 3k and v = 2i – j + 2k, show that u × v = 7i + 4j – 5k. (2)
(b)
Let w = λu + µv where λ and µ are scalars. Show that w is perpendicular to the line of intersection of the planes x + 2y + 3z = 5 and 2x – y + 2z = 7 for all values of λ and µ.
(4) (Total 6 marks)
9
15.
The position vectors of points P and Q are: p = 3i + 2j + k q = i + 3j – 2k (a)
Find the vector product p × q.
(b)
Using your answer to part (a), or otherwise, find the area of the parallelogram with two sides OP and OQ .
Working:
Answers: (a) .................................................................. (b) .................................................................. (Total 3 marks)
10
16.
Find the coordinates of the point of intersection of the line L with the plane P where:
x + 3 y –1 z –1 = = 2 –1 2 P : 2x + 3y – z = – 5 L:
Working:
Answer: ………………………………………….. (Total 3 marks)
17.
Solve, by any method, the following system of equations: 3x – 2y + z = –4 x + y –z = –2 2x + 3y = 4 Working:
Answer: ………………………………………….. (Total 3 marks)
11
18.
For the vectors a = 2i + j – 2k, b = 2i –j – k and c = i + 2j + 2k, show that: (a)
a × b = –3i – 2j – 4k (2)
(b)
(a × b) × c = –(b • c)a (3) (Total 5 marks)
19.
Three points A,
B and C have coordinates (2, 1, –2), (2, –1, –1) and (1, 2, 2) respectively. The
vectors OA , OB and OC , where O is the origin, form three concurrent edges of a parallelepiped OAPBCQSR as shown in the following diagram.
P
S
A
Q B
O (a)
R C
Find the coordinates of P, Q, R and S. (4)
(b)
Find an equation for the plane OAPB. (2)
(c)
Calculate the volume, V, of the parallelepiped given that V = OA × OB • OC ⎢ (2) (Total 8 marks)
12
20.
Find the equation of the line of intersection of the two planes –4x + y + z = –2 and 3x – y + 2z = –1. Working:
Answer: .......................................................................... (Total 3 marks)
21.
Find an equation of the plane containing the two lines
x –1=
1– y x +1 2 – y z + 2 = z – 2 and = = . 2 3 3 5
Working:
Answer: .......................................................................... (Total 3 marks)
22.
The triangle ABC has vertices at the points A(–l, 2, 3), B(–l, 3, 5) and C(0, –1, 1). (a)
Find the size of the angle θ between the vectors AB and AC . (4)
(b)
Hence, or otherwise, find the area of triangle ABC. (2)
13
Let l1 be the line parallel to AB which es through D(2, –1, 0) and l2 be the line parallel to
AC
(c)
which es through E(–l, 1, 1).
(i)
Find the equations of the lines l1 and l2.
(ii)
Hence show that l1 and l2 do not intersect. (5)
(d)
Find the shortest distance between l1 and l2. (5) (Total 16 marks)
23.
Point A(3, 0, –2) lies on the line r = 3i – 2k + λ(2i – 2j + k), where λ is a real parameter. Find the coordinates of one point which is 6 units from A, and on the line. Working:
Answer: .......................................................................... (Total 3 marks)
14
24.
(a)
Solve the following system of linear equations x + 3y – 2z = –6 2x + y + 3z = 7 3x – y + z = 6. (3)
(b)
Find the vector v = (i + 3j – 2k) × (2i + j + 3k). (3)
(c)
If a = i + 3j – 2k, b = 2i + j + 3k and u = ma + nb where m, n are scalars, and u ≠ 0, show that v is perpendicular to u for all m and n. (3)
(d)
The line l lies in the plane 3x – y + z = 6, es through the point (1, –1, 2) and is perpendicular to v. Find the equation of l. (4) (Total 13 marks)
25.
Find the angle between the vectors v = i + j + 2k and w = 2i + 3j + k. Give your answer in radians. Working:
Answer: .......................................................................... (Total 6 marks)
15
26.
The vector equations of the lines L1 and L2 are given by L1: r = i + j + k + λ(i + 2j + 3k); L2: r = i + 4j + 5k + µ(2i + j + 2k). The two lines intersect at the point P. Find the position vector of P. Working:
Answer: .......................................................................... (Total 6 marks)
27.
The points A, B, C, D have the following coordinates A : (1, 3, 1) (a)
B : (1, 2, 4)
C : (2, 3, 6)
D : (5, – 2, 1).
(i)
Evaluate the vector product AB × AC , giving your answer in of the unit vectors i, j, k.
(ii)
Find the area of the triangle ABC. (6)
The plane containing the points A, B, C is denoted by Π and the line ing through D perpendicular to Π is denoted by L. The point of intersection of L and Π is denoted by P. (b)
(i)
Find the cartesian equation of Π.
(ii)
Find the cartesian equation of L. (5)
(c)
Determine the coordinates of P. (3)
(d)
Find the perpendicular distance of D from Π. (2) (Total 16 marks)
28.
Find an equation for the line of intersection of the following two planes.
16
x + 2y – 3z = 2 2x + 3y – 5z = 3 Working:
Answer: .......................................................................... (Total 6 marks)
29.
Given two non-zero vectors a and b such that ⏐a + b⏐ = ⏐a – b⏐, find the value of a ⋅ b. Working:
Answer: .......................................................................... (Total 6 marks)
17
30.
Consider the points A (1, 3, –17) and B (6, – 7, 8) which lie on the line l. (a)
Find an equation of line l, giving the answer in parametric form. (4)
(b)
The point P is on l such that OP is perpendicular to l. Find the coordinates of P. (3) (Total 7 marks)
31.
Given that a = i + 2j – k, b = –3i + 2j + 2k and c = 2i – 3j + 4k, find (a × b) ⋅ c. Working:
Answer: ......................................................................... (Total 6 marks)
18
32.
The point A is the foot of the perpendicular from the point (1, 1, 9) to the plane 2x + y – z = 6. Find the coordinates of A. Working:
Answer: ......................................................................... (Total 6 marks)
33.
The variables x, y, z satisfy the simultaneous equations x + 2y + z = k 2x + y + 4z = 6 x – 4y + 5z = 9 where k is a constant. (a)
(i)
Show that these equations do not have a unique solution.
(ii)
Find the value of k for which the equations are consistent (that is, they can be solved). (6)
(b)
For this value of k, find the general solution of these equations. (3) (Total 9 marks)
19
34.
Consider the points A(1, 2, –4), B(l, 5, 0) and C(6, 5, –12). Find the area of ΔABC. Working:
Answer: ......................................................................... (Total 6 marks)
35.
Find the angle between the plane 3x – 2y + 4z = 12 and the z-axis. Give your answer to the nearest degree. Working:
Answer: ......................................................................... (Total 6 marks)
20
36.
The point A (2, 5, –1) is on the line L, which is perpendicular to the plane with equation x + y + z – 1 = 0. (a)
Find the Cartesian equation of the line L. (2)
(b)
Find the point of intersection of the line L and the plane. (4)
(c)
The point A is reflected in the plane. Find the coordinates of the image of A. (2)
(d)
Calculate the distance from the point B(2, 0, 6) to the line L. (4) (Total 12 marks)
37.
Consider the following system of equations where b is a constant. 3x + y + z = 1 2x + y – z = 4 5x + y + bz = 1 (a)
Solve for z in of b. (4)
(b)
Hence write down, with a reason, the range of values of b for which this system of equations has a unique solution. (2) (Total 6 marks)
21
38.
The line r = i + k + µ(i – j + 2k) and the plane 2x – y + z + 2 = 0 intersect at the point P. Find the coordinates of P. Working:
Answer: ......................................................................... (Total 6 marks)
39.
(a)
The point P(1, 2, 11) lies in the plane π1. The vector 3i – 4 j + k is perpendicular to π1. Find the Cartesian equation of π1. (2)
(b)
The plane π2 has equation x + 3y – z = –4. (i)
Show that the point P also lies in the plane π2.
(ii)
Find a vector equation of the line of intersection of π1 and π2. (5)
(c)
Find the acute angle between π1 and π2. (5) (Total 12 marks)
22
40.
Consider the four points A(1, 4, –1), B(2, 5, –2), C(5, 6, 3) and D(8, 8, 4). Find the point of intersection of the lines (AB) and (CD). Working:
Answer: ......................................................................... (Total 6 marks)
23
41.
A line l1 has equation (a)
x+2 y z −9 = = . 3 1 −2
Let M be a point on l1 with parameter µ. Express the coordinates of M in of µ. (1)
(b)
The line l2 is parallel to l1 and es through P(4, 0, –3). (i)
Write down an equation for l2.
(ii)
Express PM in of µ. (4)
(c)
The vector PM is perpendicular to l1. (i)
Find the value of µ.
(ii)
Find the distance between l1 and l2. (5)
(d)
The plane π1 contains l1 and l2. Find an equation for π1, giving your answer in the form Ax + By + Cz = D. (4)
(e)
The plane π2 has equation x – 5y – z = –11. that l1 is the line of intersection of the planes π1 and π2. (2) (Total 16 marks)
24
42.
A triangle has its vertices at A(–1, 3, 2), B(3, 6, 1) and C(–4, 4, 3). (a)
Show that AB • AC = –10.
(b)
ˆ C = –0.591. Show that, to three significant figures, cos BA ..................................................................................................................................... ..................................................................................................................................... ..................................................................................................................................... ..................................................................................................................................... ..................................................................................................................................... ..................................................................................................................................... ..................................................................................................................................... ..................................................................................................................................... ..................................................................................................................................... ..................................................................................................................................... ..................................................................................................................................... ..................................................................................................................................... (Total 6 marks)
25
43.
(a)
Write down the inverse of the matrix
⎛ 1 − 3 1 ⎞ ⎜ ⎟ A = ⎜ 2 2 − 1⎟ ⎜ 1 − 5 3 ⎟ ⎝ ⎠ (b)
Hence, find the point of intersection of the three planes. x – 3y + z = 1 2x + 2y – z = 2 x – 5y + 3z = 3
(c)
A fourth plane with equation x + y + z = d es through the point of intersection. Find the value of d. ..................................................................................................................................... ..................................................................................................................................... ..................................................................................................................................... ..................................................................................................................................... ..................................................................................................................................... ..................................................................................................................................... ..................................................................................................................................... ..................................................................................................................................... ..................................................................................................................................... ..................................................................................................................................... ..................................................................................................................................... ..................................................................................................................................... (Total 6 marks)
26
44.
(a)
x−2 y−2 z −3 x−2 y−3 z −4 = = = = and intersect and find the 1 3 1 1 4 2 coordinates of P, the point of intersection. Show that lines
(8)
(b) (c)
Find the Cartesian equation of the plane ∏ that contains the two lines.
(6)
The point q (3, 4, 3) lies on ∏. The line L es through the midpoint of [PQ]. Point S is on L such that PS = QS = 3 , and the triangle PQS is normal to the plane ∏. Given that there are two possible positions for S, find their coordinates. (15) (Total 29 marks)
45.
⎛ 2 ⎞ ⎛ 2 ⎞ ⎜ ⎟ ⎜ ⎟ The position vectors of points P and Q are ⎜ − 3 ⎟ and ⎜ 2 ⎟ respectively. The origin is at O. ⎜ 1 ⎟ ⎜ − 4 ⎟ ⎝ ⎠ ⎝ ⎠ Find (a)
ˆQ; the angle PO
(b)
the area of the triangle OPQ.
Working:
Answers: (a) (b) (Total 6 marks)
46.
(a)
⎛ 2 ⎞ ⎛ − 2 ⎞ ⎛ 1 ⎞ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ The plane π1 has equation r = ⎜ 1 ⎟ + λ ⎜ 1 ⎟ + µ ⎜ − 3 ⎟ . ⎜ 1 ⎟ ⎜ 8 ⎟ ⎜ − 9 ⎟ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠
27
⎛ 2 ⎞ ⎛ 1 ⎞ ⎛1⎞ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ The plane π2 has the equation r = ⎜ 0 ⎟ + s⎜ 2 ⎟ + t ⎜1⎟ . ⎜ 1 ⎟ ⎜ 1 ⎟ ⎜1⎟ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ (i)
For points which lie in π1 and π2, show that, λ = µ.
(ii)
Hence, or otherwise, find a vector equation of the line of intersection of π and π2. (5)
(b)
y The plane π3 contains the line 2 − x = = z + 1 and is perpendicular to 3i – 2j + k. 3 −4 Find the cartesian equation of π3. (4)
(c)
Find the intersection of π1, π2 and π3. (3) (Total 12 marks)
47.
The parallelogram ABCD has vertices A (3, 2, 0), B (7, –1, 1), C (10, –3, 0) and D (6, 0, 1). Calculate the area of the parallelogram.
Working:
Answer: (Total 6 marks)
28
48.
The line
y−2 z+2 x −1 = = is reflected in the plane x + y + z = 1. Calculate the angle 1 1 1
between the line and its reflection. Give your answer in radians.
Working:
Answer: (Total 6 marks)
49.
The plane π contains the line (a) (b)
50.
Show that the equation of π is 6x + 2y – 3z = –7. Calculate the distance of the plane π from the origin.
⎛ 2 ⎞ ⎜ ⎟ Let a = ⎜ 1 ⎟ , b = ⎜ 0 ⎟ ⎝ ⎠ (a)
y −1 z −5 x −1 = = and the point (1, −2, 3). 6 3 2
⎛ − 1⎞ ⎜ ⎟ ⎜ p ⎟ and c = ⎜ 6 ⎟ ⎝ ⎠
(7)
(4) (Total 11 marks)
⎛ 2 ⎞ ⎜ ⎟ ⎜ − 4 ⎟ . ⎜ 3 ⎟ ⎝ ⎠
Find a × b.
29
(b)
Find the value of p, given that a × b is parallel to c.
.............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. (Total 6 marks)
51.
Let A be the point (2, –1, 0), B the point (3, 0, 1) and C the point (1, m, 2), where m∈ 0. (a)
(i)
Find the scalar product BA • BC .
(ii)
ˆ C = arccos Hence, given that AB
,m<
2 , show that m = –1. 3 (6)
(b)
Determine the Cartesian equation of the plane ABC. (4)
(c)
Find the area of triangle ABC. (3)
(d)
(i)
The line L is perpendicular to plane ABC and es through A. Find a vector equation of L.
(ii)
The point D(6, –7, 2) lies on L. Find the volume of the pyramid ABCD. (8) (Total 21 marks)
52.
The lines L1 and L2 have parametric equations 30
L1 : x = 1 + 2λ, y = 1 + 3λ, z = 1 − λ L2 : x = 2 – µ, y = 3 + 4µ, z = 4 + 2µ Find the angle between L1 and L2. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. (Total 6 marks)
31
53.
Let P be the point (1, 0, – 2) and Π be the plane x + y − 2z + 3 = 0. Let P′ be the reflection of P in the plane Π. Find the coordinates of the point P′. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. (Total 6 marks)
54.
(a)
The line l1 es through the point A (0, 1, 2) and is perpendicular to the plane x − 4y − 3z = 0. Find the Cartesian equations of l1. (2)
(b)
The line l2 is parallel to l1 and es through the point P(3, –8, –11). Find the vector equation of the line l2. (2)
(c)
(i)
The point Q is on the line l1 such that PQ is perpendicular to l1 and l2. Find the coordinates of Q.
(ii)
Hence find the distance between l1 and l2. (10) (Total 14 marks)
32
55.
Consider the system of equations x + 2y + kz = 0 x + 3y + z = 3 kx + 8y + 5z = 6 (a)
Find the set of values of k for which this system of equations has a unique solution. (6)
(b)
For each value of k that results in a non-unique solution, find the solution set. (8) (Total 14 marks)
33
56.
Consider the vectors a, b, c, d
⎛ 2 ⎞ ⎛ 1 ⎞ ⎜ ⎟ ⎜ ⎟ a = ⎜ 3 ⎟ , b = ⎜ 2 ⎟ , c ⎜ − 1⎟ ⎜ 5 ⎟ ⎝ ⎠ ⎝ ⎠
⎛ 3 ⎞ ⎛ µ ⎞ ⎜ ⎟ ⎜ ⎟ = ⎜ 1 ⎟ , d = ⎜ − 2 ⎟. ⎜ λ ⎟ ⎜ 15 ⎟ ⎝ ⎠ ⎝ ⎠
Let s = (a • b) c + d, where s is perpendicular to a. Find an expression for λ in of µ. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. (Total 6 marks)
34
57.
Find the cosine of the angle θ between the planes π1 and π2, where π1 has equation −2x + y − z = 2 and π2 has equation x + 2y − z = 6. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. (Total 6 marks)
35
58.
Given that a = 2i − j − k, b = 2i + j − 2k and c = −i + j − k are the position vectors of the points A, B and C respectively, calculate the area of triangle ABC. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. (Total 6 marks)
36
59.
Two planes π1 and π2 are represented by the equations
⎛ 3 ⎞ ⎛ − 2 ⎞ ⎛ 2 ⎞ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ π1: r = ⎜ 1 ⎟ + λ ⎜ 2 ⎟ + µ⎜ 1 ⎟ ⎜ 5 ⎟ ⎜ 3 ⎟ ⎜ 0 ⎟ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ π2: 2x – y – 2z = 4.
(a)
(i)
⎛ − 2 ⎞ ⎛ 2 ⎞ ⎜ ⎟ ⎜ ⎟ Find ⎜ 2 ⎟ × ⎜ 1 ⎟. ⎜ 3 ⎟ ⎜ 0 ⎟ ⎝ ⎠ ⎝ ⎠
(ii)
Show that the equation of π1 can be written as x − 2y + 2z =11. (4)
(b)
Show that π1 is perpendicular to π2. (4)
(c)
The line l1 is the line of intersection of π1 and π2. Find the vector equation of l1, giving the answer in parametric form. (5)
(d)
The line l2 is parallel to both π1 and π2, and es through P(3, –5, –1). Find an equation for l2 in Cartesian form. (3)
(e)
Let Q be the foot of the perpendicular from P to the plane π2. (i)
Find the coordinates of Q.
(ii)
Find PQ. (7) (Total 23 marks)
37
60.
Consider the vectors a = i − j + k, b = i + 2 j + 4k and c = 2i − 5 j − k. (a)
Given that c = ma + nb where m, n∈
, find the value of m and of n. (5)
(b)
Find a unit vector, u, normal to both a and b. (5)
(c)
The plane π1 contains the point A (1, –1, 1) and is normal to b. The plane intersects the x, y and z axes at the points L, M and N respectively. (i)
Find a Cartesian equation of π1.
(ii)
Write down the coordinates of L, M and N. (5)
(d)
The line through the origin, O, normal to π1 meets π1 at the point P. (i)
Find the coordinates of P.
(ii)
Hence find the distance of π1 from the origin. (7)
(e)
The plane π2 has equation x + 2y + 4z = 4. Calculate the angle between π2 and a line parallel to a. (5) (Total 27 marks)
38
61.
Find the non-unique solution for the following system of simultaneous equations x−y−z=3 x − 2y + z = 2 2x − y − 4z = 7 .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. (Total 6 marks)
39
62.
The lines l1 and l2 have equations
⎛ 4 ⎞ ⎛ 1 ⎞ ⎜ ⎟ ⎜ ⎟ r1 = ⎜ 3 ⎟ + λ ⎜ 5 ⎟ and r2 = ⎜ 0 ⎟ ⎜ − 2 ⎟ ⎝ ⎠ ⎝ ⎠
⎛ 2 ⎞ ⎛ 0 ⎞ ⎜ ⎟ ⎜ ⎟ ⎜ − 1⎟ + µ ⎜ 2 ⎟ ⎜ 3 ⎟ ⎜ − 3 ⎟ ⎝ ⎠ ⎝ ⎠
respectively, where λ and µ are parameters. (a)
Show that l1 es through the point (2, − 7, 4). (2)
(b)
Determine whether the lines l1 and l2 intersect.
.............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. (4) (Total 6 marks)
40
63.
⎛ 2 ⎞ ⎜ ⎟ x−4 y +2 z −6 = = . A plane Π has equation r • ⎜ − 1⎟ = 16 and a line l has equations −1 2 4 ⎜ 1 ⎟ ⎝ ⎠ Show that the line l lies in the plane Π. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. (Total 6 marks)
41
64.
The points A, B, C have position vectors i + j + 2k, i + 2 j + k, i + k respectively and lie in the plane π. (a)
Find (i)
the area of the triangle ABC;
(ii)
the shortest distance from C to the line AB;
(iii)
the cartesian equation of the plane π.
(14)
The line L es through the origin and is normal to the plane π, it intersects π at the point D. (b)
Find (i)
the coordinates of the point D;
(ii)
the distance of π from the origin.
(6) (Total 20 marks)
42
65.
Given any two non-zero vectors a and b, show that
2
a ×b = a
2
2
2
b − (a • b) .
.............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. (Total 6 marks)
66.
Consider the points A(1, −1, 4), B(2, − 2, 5) and O(0, 0, 0). (a)
Calculate the cosine of the angle between OA and AB. (5)
(b)
Find a vector equation of the line L1 which es through A and B. (2)
The line L2 has equation r = 2i + j + 7k + t(2i + j + 3k), where t∈ . (c)
Show that the lines L1 and L2 intersect and find the coordinates of their point of intersection. (7)
(d)
Find the Cartesian equation of the plane which contains both the line L2 and the point A. (6) (Total 20 marks)
67.
⎛ 4 ⎞ ⎜ ⎟ A ray of light coming from the point (−1, 3, 2) is travelling in the direction of vector ⎜ 1 ⎟ ⎜ − 2 ⎟ ⎝ ⎠ 43
and meets the plane π : x + y + 2z − 24 = 0. Find the angle that the ray of light makes with the plane. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. (Total 6 marks)
44
68.
Find the vector equation of the line of intersection of the three planes represented by the following system of equations. 2x − 7y + 5z =1 6x + 3y – z = –1 −14x − 23y +13z = 5 .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. (Total 6 marks)
45
Find the coordinates of the point where the line given by the parametric equations x = 2λ + 4, y = –λ – 2, z = 3λ + 2, intersects the plane with equation 2x + 3y – z = 2. Working:
Answer: ………………………………………….. (Total 4 marks)
2.
Find the value of a for which the following system of equations does not have a unique solution. 4x – y + 2z = 1 2x + 3y
= –6
x – 2y + az = 7 2 Working:
Answer: ………………………………………….. (Total 4 marks)
1
3.
(a)
Find a vector perpendicular to the two vectors:
!
!
!
OP = i – 3 j + 2 k
! ! ! OQ = –2 i + j – k (b)
If OP and OQ are position vectors for the points P and Q, use your answer to part (a), or otherwise, to find the area of the triangle OPQ.
Working:
Answers: (a) ………………………………………….. (b) …………………………………….......... (Total 4 marks)
2
4.
The coordinates of the points P, Q, R and S are (4, 1, –1), (3, 3, 5), (1, 0, 2c), and (1, 1, 2), respectively. (a)
Find the value of c so that the vectors QR and PR are orthogonal. (7)
For the remainder of the question, use the value of c found in part (a) for the coordinate of the point R. (b)
Evaluate PS × PR . (4)
(c)
Find an equation of the line l which es through the point Q and is parallel to the vector PR . (3)
(d) (e)
5.
Find an equation of the plane π which contains the line l and es through the point S. Find the shortest distance between the point P and the plane π.
(4)
(4) (Total 22 marks)
! ! ! ! The vector n = 2 i – j +3 k is normal to a plane which es through the point (2, 1, 2). (a)
Find an equation for the plane.
(b)
Find a if the point (a, a – 1, a – 2) lies on the plane.
Working:
Answers: (a) .................................................................. (b) .................................................................. (Total 4 marks)
6.
The rectangle box shown in the diagram has dimensions 6 cm × 4 cm × 3 cm.
3
H
G
E
F
3cm
D A
C 6cm
B
4cm
Find, correct to the nearest one-tenth of a degree, the size of the angle AHˆ C . Working:
Answer: ....…………………………………….......... (Total 4 marks)
4
7.
Calculate the shortest distance from the point A(0, 2, 2) to the line
! ! ! ! ! ! ! r = 5 i + 9 j + 6 k + t( i + 2 j + 2 k ) where t is a scalar. Working:
Answer: ....…………………………………….......... (Total 4 marks)
5
8.
Consider the points A(l, 2, 1), B(0, –1, 2), C(1, 0, 2), and D(2, –1, –6). (a)
Find the vectors AB and BC . (2)
(b)
Calculate AB × BC . (3)
(c)
Hence, or otherwise find the area of triangle ABC. (2)
(d)
Find the equation of the plane P containing the points A, B, and C. (3)
(e)
Find a set of parametric equations for the line through the point D and perpendicular to the plane P. (2)
(f)
Find the distance from the point D to the plane P. (3)
(g)
Find a unit vector which is perpendicular to the plane P. (2)
(h)
The point E is a reflection of D in the plane P. Find the coordinates of E. (4) (Total 21 marks)
6
9.
The system of equations represented by the following matrix equation has an infinite number of solutions.
⎛ 2 ⎜ ⎜ 1 ⎜ 2 ⎝
−1 2 1
− 9 ⎞ ⎛ x ⎞ ⎛ 7 ⎞ ⎟ ⎜ ⎟ ⎜ ⎟ 3 ⎟ ⎜ y ⎟ = ⎜ 1 ⎟ − 3 ⎟⎠ ⎜⎝ z ⎟⎠ ⎜⎝ k ⎟⎠
Find the value of k. Working:
Answer: .................................................................. (Total 3 marks)
10.
Find a vector that is normal to the plane containing the lines L1, and L2, whose equations are: L1: r = i + k + λ (2i + j – 2k) L2: r = 3i + 2j + 2k + µ (j + 3k) Working:
Answer: .................................................................. (Total 3 marks)
11.
The plane 6x – 2y + z = 11 contains the line x – 1 =
y +1 z − 3 . Find l. = 2 l
7
Working:
Answer: .................................................................. (Total 3 marks)
12.
Let a be the angle between the vectors a and b, where a = (cos θ)i + (sin θ)j, b = (sin θ)i + (cos θ)j and 0 < θ <
π . 4
Express α in of θ. Working:
Answer: .................................................................. (Total 3 marks)
8
13.
(a)
(b)
Given matrices A, B, C for which AB = C and det A ≠ 0, express B in of A and C.
⎛ 1 ⎜ Let A = ⎜ 2 ⎜ 3 ⎝
2 −1 −3
3 ⎞ ⎟ 2 ⎟ , D = 2 ⎟⎠
(i)
Find the matrix DA;
(ii)
Find B if AB = C.
⎛ − 4 13 ⎜ 7 ⎜ − 2 ⎜ 3 − 9 ⎝
− 7 ⎞ ⎟ − 4 ⎟ and C = 5 ⎟⎠
(2)
⎛ 5 ⎞ ⎜ ⎟ ⎜ 7 ⎟ . ⎜10 ⎟ ⎝ ⎠
(3)
(c)
Find the coordinates of the point of intersection of the planes x + 2y + 3z = 5, 2x – y + 2z = 7 and 3x – 3y + 2z = 10. (2) (Total 7 marks)
14.
(a)
If u = i +2j + 3k and v = 2i – j + 2k, show that u × v = 7i + 4j – 5k. (2)
(b)
Let w = λu + µv where λ and µ are scalars. Show that w is perpendicular to the line of intersection of the planes x + 2y + 3z = 5 and 2x – y + 2z = 7 for all values of λ and µ.
(4) (Total 6 marks)
9
15.
The position vectors of points P and Q are: p = 3i + 2j + k q = i + 3j – 2k (a)
Find the vector product p × q.
(b)
Using your answer to part (a), or otherwise, find the area of the parallelogram with two sides OP and OQ .
Working:
Answers: (a) .................................................................. (b) .................................................................. (Total 3 marks)
10
16.
Find the coordinates of the point of intersection of the line L with the plane P where:
x + 3 y –1 z –1 = = 2 –1 2 P : 2x + 3y – z = – 5 L:
Working:
Answer: ………………………………………….. (Total 3 marks)
17.
Solve, by any method, the following system of equations: 3x – 2y + z = –4 x + y –z = –2 2x + 3y = 4 Working:
Answer: ………………………………………….. (Total 3 marks)
11
18.
For the vectors a = 2i + j – 2k, b = 2i –j – k and c = i + 2j + 2k, show that: (a)
a × b = –3i – 2j – 4k (2)
(b)
(a × b) × c = –(b • c)a (3) (Total 5 marks)
19.
Three points A,
B and C have coordinates (2, 1, –2), (2, –1, –1) and (1, 2, 2) respectively. The
vectors OA , OB and OC , where O is the origin, form three concurrent edges of a parallelepiped OAPBCQSR as shown in the following diagram.
P
S
A
Q B
O (a)
R C
Find the coordinates of P, Q, R and S. (4)
(b)
Find an equation for the plane OAPB. (2)
(c)
Calculate the volume, V, of the parallelepiped given that V = OA × OB • OC ⎢ (2) (Total 8 marks)
12
20.
Find the equation of the line of intersection of the two planes –4x + y + z = –2 and 3x – y + 2z = –1. Working:
Answer: .......................................................................... (Total 3 marks)
21.
Find an equation of the plane containing the two lines
x –1=
1– y x +1 2 – y z + 2 = z – 2 and = = . 2 3 3 5
Working:
Answer: .......................................................................... (Total 3 marks)
22.
The triangle ABC has vertices at the points A(–l, 2, 3), B(–l, 3, 5) and C(0, –1, 1). (a)
Find the size of the angle θ between the vectors AB and AC . (4)
(b)
Hence, or otherwise, find the area of triangle ABC. (2)
13
Let l1 be the line parallel to AB which es through D(2, –1, 0) and l2 be the line parallel to
AC
(c)
which es through E(–l, 1, 1).
(i)
Find the equations of the lines l1 and l2.
(ii)
Hence show that l1 and l2 do not intersect. (5)
(d)
Find the shortest distance between l1 and l2. (5) (Total 16 marks)
23.
Point A(3, 0, –2) lies on the line r = 3i – 2k + λ(2i – 2j + k), where λ is a real parameter. Find the coordinates of one point which is 6 units from A, and on the line. Working:
Answer: .......................................................................... (Total 3 marks)
14
24.
(a)
Solve the following system of linear equations x + 3y – 2z = –6 2x + y + 3z = 7 3x – y + z = 6. (3)
(b)
Find the vector v = (i + 3j – 2k) × (2i + j + 3k). (3)
(c)
If a = i + 3j – 2k, b = 2i + j + 3k and u = ma + nb where m, n are scalars, and u ≠ 0, show that v is perpendicular to u for all m and n. (3)
(d)
The line l lies in the plane 3x – y + z = 6, es through the point (1, –1, 2) and is perpendicular to v. Find the equation of l. (4) (Total 13 marks)
25.
Find the angle between the vectors v = i + j + 2k and w = 2i + 3j + k. Give your answer in radians. Working:
Answer: .......................................................................... (Total 6 marks)
15
26.
The vector equations of the lines L1 and L2 are given by L1: r = i + j + k + λ(i + 2j + 3k); L2: r = i + 4j + 5k + µ(2i + j + 2k). The two lines intersect at the point P. Find the position vector of P. Working:
Answer: .......................................................................... (Total 6 marks)
27.
The points A, B, C, D have the following coordinates A : (1, 3, 1) (a)
B : (1, 2, 4)
C : (2, 3, 6)
D : (5, – 2, 1).
(i)
Evaluate the vector product AB × AC , giving your answer in of the unit vectors i, j, k.
(ii)
Find the area of the triangle ABC. (6)
The plane containing the points A, B, C is denoted by Π and the line ing through D perpendicular to Π is denoted by L. The point of intersection of L and Π is denoted by P. (b)
(i)
Find the cartesian equation of Π.
(ii)
Find the cartesian equation of L. (5)
(c)
Determine the coordinates of P. (3)
(d)
Find the perpendicular distance of D from Π. (2) (Total 16 marks)
28.
Find an equation for the line of intersection of the following two planes.
16
x + 2y – 3z = 2 2x + 3y – 5z = 3 Working:
Answer: .......................................................................... (Total 6 marks)
29.
Given two non-zero vectors a and b such that ⏐a + b⏐ = ⏐a – b⏐, find the value of a ⋅ b. Working:
Answer: .......................................................................... (Total 6 marks)
17
30.
Consider the points A (1, 3, –17) and B (6, – 7, 8) which lie on the line l. (a)
Find an equation of line l, giving the answer in parametric form. (4)
(b)
The point P is on l such that OP is perpendicular to l. Find the coordinates of P. (3) (Total 7 marks)
31.
Given that a = i + 2j – k, b = –3i + 2j + 2k and c = 2i – 3j + 4k, find (a × b) ⋅ c. Working:
Answer: ......................................................................... (Total 6 marks)
18
32.
The point A is the foot of the perpendicular from the point (1, 1, 9) to the plane 2x + y – z = 6. Find the coordinates of A. Working:
Answer: ......................................................................... (Total 6 marks)
33.
The variables x, y, z satisfy the simultaneous equations x + 2y + z = k 2x + y + 4z = 6 x – 4y + 5z = 9 where k is a constant. (a)
(i)
Show that these equations do not have a unique solution.
(ii)
Find the value of k for which the equations are consistent (that is, they can be solved). (6)
(b)
For this value of k, find the general solution of these equations. (3) (Total 9 marks)
19
34.
Consider the points A(1, 2, –4), B(l, 5, 0) and C(6, 5, –12). Find the area of ΔABC. Working:
Answer: ......................................................................... (Total 6 marks)
35.
Find the angle between the plane 3x – 2y + 4z = 12 and the z-axis. Give your answer to the nearest degree. Working:
Answer: ......................................................................... (Total 6 marks)
20
36.
The point A (2, 5, –1) is on the line L, which is perpendicular to the plane with equation x + y + z – 1 = 0. (a)
Find the Cartesian equation of the line L. (2)
(b)
Find the point of intersection of the line L and the plane. (4)
(c)
The point A is reflected in the plane. Find the coordinates of the image of A. (2)
(d)
Calculate the distance from the point B(2, 0, 6) to the line L. (4) (Total 12 marks)
37.
Consider the following system of equations where b is a constant. 3x + y + z = 1 2x + y – z = 4 5x + y + bz = 1 (a)
Solve for z in of b. (4)
(b)
Hence write down, with a reason, the range of values of b for which this system of equations has a unique solution. (2) (Total 6 marks)
21
38.
The line r = i + k + µ(i – j + 2k) and the plane 2x – y + z + 2 = 0 intersect at the point P. Find the coordinates of P. Working:
Answer: ......................................................................... (Total 6 marks)
39.
(a)
The point P(1, 2, 11) lies in the plane π1. The vector 3i – 4 j + k is perpendicular to π1. Find the Cartesian equation of π1. (2)
(b)
The plane π2 has equation x + 3y – z = –4. (i)
Show that the point P also lies in the plane π2.
(ii)
Find a vector equation of the line of intersection of π1 and π2. (5)
(c)
Find the acute angle between π1 and π2. (5) (Total 12 marks)
22
40.
Consider the four points A(1, 4, –1), B(2, 5, –2), C(5, 6, 3) and D(8, 8, 4). Find the point of intersection of the lines (AB) and (CD). Working:
Answer: ......................................................................... (Total 6 marks)
23
41.
A line l1 has equation (a)
x+2 y z −9 = = . 3 1 −2
Let M be a point on l1 with parameter µ. Express the coordinates of M in of µ. (1)
(b)
The line l2 is parallel to l1 and es through P(4, 0, –3). (i)
Write down an equation for l2.
(ii)
Express PM in of µ. (4)
(c)
The vector PM is perpendicular to l1. (i)
Find the value of µ.
(ii)
Find the distance between l1 and l2. (5)
(d)
The plane π1 contains l1 and l2. Find an equation for π1, giving your answer in the form Ax + By + Cz = D. (4)
(e)
The plane π2 has equation x – 5y – z = –11. that l1 is the line of intersection of the planes π1 and π2. (2) (Total 16 marks)
24
42.
A triangle has its vertices at A(–1, 3, 2), B(3, 6, 1) and C(–4, 4, 3). (a)
Show that AB • AC = –10.
(b)
ˆ C = –0.591. Show that, to three significant figures, cos BA ..................................................................................................................................... ..................................................................................................................................... ..................................................................................................................................... ..................................................................................................................................... ..................................................................................................................................... ..................................................................................................................................... ..................................................................................................................................... ..................................................................................................................................... ..................................................................................................................................... ..................................................................................................................................... ..................................................................................................................................... ..................................................................................................................................... (Total 6 marks)
25
43.
(a)
Write down the inverse of the matrix
⎛ 1 − 3 1 ⎞ ⎜ ⎟ A = ⎜ 2 2 − 1⎟ ⎜ 1 − 5 3 ⎟ ⎝ ⎠ (b)
Hence, find the point of intersection of the three planes. x – 3y + z = 1 2x + 2y – z = 2 x – 5y + 3z = 3
(c)
A fourth plane with equation x + y + z = d es through the point of intersection. Find the value of d. ..................................................................................................................................... ..................................................................................................................................... ..................................................................................................................................... ..................................................................................................................................... ..................................................................................................................................... ..................................................................................................................................... ..................................................................................................................................... ..................................................................................................................................... ..................................................................................................................................... ..................................................................................................................................... ..................................................................................................................................... ..................................................................................................................................... (Total 6 marks)
26
44.
(a)
x−2 y−2 z −3 x−2 y−3 z −4 = = = = and intersect and find the 1 3 1 1 4 2 coordinates of P, the point of intersection. Show that lines
(8)
(b) (c)
Find the Cartesian equation of the plane ∏ that contains the two lines.
(6)
The point q (3, 4, 3) lies on ∏. The line L es through the midpoint of [PQ]. Point S is on L such that PS = QS = 3 , and the triangle PQS is normal to the plane ∏. Given that there are two possible positions for S, find their coordinates. (15) (Total 29 marks)
45.
⎛ 2 ⎞ ⎛ 2 ⎞ ⎜ ⎟ ⎜ ⎟ The position vectors of points P and Q are ⎜ − 3 ⎟ and ⎜ 2 ⎟ respectively. The origin is at O. ⎜ 1 ⎟ ⎜ − 4 ⎟ ⎝ ⎠ ⎝ ⎠ Find (a)
ˆQ; the angle PO
(b)
the area of the triangle OPQ.
Working:
Answers: (a) (b) (Total 6 marks)
46.
(a)
⎛ 2 ⎞ ⎛ − 2 ⎞ ⎛ 1 ⎞ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ The plane π1 has equation r = ⎜ 1 ⎟ + λ ⎜ 1 ⎟ + µ ⎜ − 3 ⎟ . ⎜ 1 ⎟ ⎜ 8 ⎟ ⎜ − 9 ⎟ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠
27
⎛ 2 ⎞ ⎛ 1 ⎞ ⎛1⎞ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ The plane π2 has the equation r = ⎜ 0 ⎟ + s⎜ 2 ⎟ + t ⎜1⎟ . ⎜ 1 ⎟ ⎜ 1 ⎟ ⎜1⎟ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ (i)
For points which lie in π1 and π2, show that, λ = µ.
(ii)
Hence, or otherwise, find a vector equation of the line of intersection of π and π2. (5)
(b)
y The plane π3 contains the line 2 − x = = z + 1 and is perpendicular to 3i – 2j + k. 3 −4 Find the cartesian equation of π3. (4)
(c)
Find the intersection of π1, π2 and π3. (3) (Total 12 marks)
47.
The parallelogram ABCD has vertices A (3, 2, 0), B (7, –1, 1), C (10, –3, 0) and D (6, 0, 1). Calculate the area of the parallelogram.
Working:
Answer: (Total 6 marks)
28
48.
The line
y−2 z+2 x −1 = = is reflected in the plane x + y + z = 1. Calculate the angle 1 1 1
between the line and its reflection. Give your answer in radians.
Working:
Answer: (Total 6 marks)
49.
The plane π contains the line (a) (b)
50.
Show that the equation of π is 6x + 2y – 3z = –7. Calculate the distance of the plane π from the origin.
⎛ 2 ⎞ ⎜ ⎟ Let a = ⎜ 1 ⎟ , b = ⎜ 0 ⎟ ⎝ ⎠ (a)
y −1 z −5 x −1 = = and the point (1, −2, 3). 6 3 2
⎛ − 1⎞ ⎜ ⎟ ⎜ p ⎟ and c = ⎜ 6 ⎟ ⎝ ⎠
(7)
(4) (Total 11 marks)
⎛ 2 ⎞ ⎜ ⎟ ⎜ − 4 ⎟ . ⎜ 3 ⎟ ⎝ ⎠
Find a × b.
29
(b)
Find the value of p, given that a × b is parallel to c.
.............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. (Total 6 marks)
51.
Let A be the point (2, –1, 0), B the point (3, 0, 1) and C the point (1, m, 2), where m∈ 0. (a)
(i)
Find the scalar product BA • BC .
(ii)
ˆ C = arccos Hence, given that AB
,m<
2 , show that m = –1. 3 (6)
(b)
Determine the Cartesian equation of the plane ABC. (4)
(c)
Find the area of triangle ABC. (3)
(d)
(i)
The line L is perpendicular to plane ABC and es through A. Find a vector equation of L.
(ii)
The point D(6, –7, 2) lies on L. Find the volume of the pyramid ABCD. (8) (Total 21 marks)
52.
The lines L1 and L2 have parametric equations 30
L1 : x = 1 + 2λ, y = 1 + 3λ, z = 1 − λ L2 : x = 2 – µ, y = 3 + 4µ, z = 4 + 2µ Find the angle between L1 and L2. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. (Total 6 marks)
31
53.
Let P be the point (1, 0, – 2) and Π be the plane x + y − 2z + 3 = 0. Let P′ be the reflection of P in the plane Π. Find the coordinates of the point P′. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. (Total 6 marks)
54.
(a)
The line l1 es through the point A (0, 1, 2) and is perpendicular to the plane x − 4y − 3z = 0. Find the Cartesian equations of l1. (2)
(b)
The line l2 is parallel to l1 and es through the point P(3, –8, –11). Find the vector equation of the line l2. (2)
(c)
(i)
The point Q is on the line l1 such that PQ is perpendicular to l1 and l2. Find the coordinates of Q.
(ii)
Hence find the distance between l1 and l2. (10) (Total 14 marks)
32
55.
Consider the system of equations x + 2y + kz = 0 x + 3y + z = 3 kx + 8y + 5z = 6 (a)
Find the set of values of k for which this system of equations has a unique solution. (6)
(b)
For each value of k that results in a non-unique solution, find the solution set. (8) (Total 14 marks)
33
56.
Consider the vectors a, b, c, d
⎛ 2 ⎞ ⎛ 1 ⎞ ⎜ ⎟ ⎜ ⎟ a = ⎜ 3 ⎟ , b = ⎜ 2 ⎟ , c ⎜ − 1⎟ ⎜ 5 ⎟ ⎝ ⎠ ⎝ ⎠
⎛ 3 ⎞ ⎛ µ ⎞ ⎜ ⎟ ⎜ ⎟ = ⎜ 1 ⎟ , d = ⎜ − 2 ⎟. ⎜ λ ⎟ ⎜ 15 ⎟ ⎝ ⎠ ⎝ ⎠
Let s = (a • b) c + d, where s is perpendicular to a. Find an expression for λ in of µ. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. (Total 6 marks)
34
57.
Find the cosine of the angle θ between the planes π1 and π2, where π1 has equation −2x + y − z = 2 and π2 has equation x + 2y − z = 6. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. (Total 6 marks)
35
58.
Given that a = 2i − j − k, b = 2i + j − 2k and c = −i + j − k are the position vectors of the points A, B and C respectively, calculate the area of triangle ABC. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. (Total 6 marks)
36
59.
Two planes π1 and π2 are represented by the equations
⎛ 3 ⎞ ⎛ − 2 ⎞ ⎛ 2 ⎞ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ π1: r = ⎜ 1 ⎟ + λ ⎜ 2 ⎟ + µ⎜ 1 ⎟ ⎜ 5 ⎟ ⎜ 3 ⎟ ⎜ 0 ⎟ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ π2: 2x – y – 2z = 4.
(a)
(i)
⎛ − 2 ⎞ ⎛ 2 ⎞ ⎜ ⎟ ⎜ ⎟ Find ⎜ 2 ⎟ × ⎜ 1 ⎟. ⎜ 3 ⎟ ⎜ 0 ⎟ ⎝ ⎠ ⎝ ⎠
(ii)
Show that the equation of π1 can be written as x − 2y + 2z =11. (4)
(b)
Show that π1 is perpendicular to π2. (4)
(c)
The line l1 is the line of intersection of π1 and π2. Find the vector equation of l1, giving the answer in parametric form. (5)
(d)
The line l2 is parallel to both π1 and π2, and es through P(3, –5, –1). Find an equation for l2 in Cartesian form. (3)
(e)
Let Q be the foot of the perpendicular from P to the plane π2. (i)
Find the coordinates of Q.
(ii)
Find PQ. (7) (Total 23 marks)
37
60.
Consider the vectors a = i − j + k, b = i + 2 j + 4k and c = 2i − 5 j − k. (a)
Given that c = ma + nb where m, n∈
, find the value of m and of n. (5)
(b)
Find a unit vector, u, normal to both a and b. (5)
(c)
The plane π1 contains the point A (1, –1, 1) and is normal to b. The plane intersects the x, y and z axes at the points L, M and N respectively. (i)
Find a Cartesian equation of π1.
(ii)
Write down the coordinates of L, M and N. (5)
(d)
The line through the origin, O, normal to π1 meets π1 at the point P. (i)
Find the coordinates of P.
(ii)
Hence find the distance of π1 from the origin. (7)
(e)
The plane π2 has equation x + 2y + 4z = 4. Calculate the angle between π2 and a line parallel to a. (5) (Total 27 marks)
38
61.
Find the non-unique solution for the following system of simultaneous equations x−y−z=3 x − 2y + z = 2 2x − y − 4z = 7 .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. (Total 6 marks)
39
62.
The lines l1 and l2 have equations
⎛ 4 ⎞ ⎛ 1 ⎞ ⎜ ⎟ ⎜ ⎟ r1 = ⎜ 3 ⎟ + λ ⎜ 5 ⎟ and r2 = ⎜ 0 ⎟ ⎜ − 2 ⎟ ⎝ ⎠ ⎝ ⎠
⎛ 2 ⎞ ⎛ 0 ⎞ ⎜ ⎟ ⎜ ⎟ ⎜ − 1⎟ + µ ⎜ 2 ⎟ ⎜ 3 ⎟ ⎜ − 3 ⎟ ⎝ ⎠ ⎝ ⎠
respectively, where λ and µ are parameters. (a)
Show that l1 es through the point (2, − 7, 4). (2)
(b)
Determine whether the lines l1 and l2 intersect.
.............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. (4) (Total 6 marks)
40
63.
⎛ 2 ⎞ ⎜ ⎟ x−4 y +2 z −6 = = . A plane Π has equation r • ⎜ − 1⎟ = 16 and a line l has equations −1 2 4 ⎜ 1 ⎟ ⎝ ⎠ Show that the line l lies in the plane Π. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. (Total 6 marks)
41
64.
The points A, B, C have position vectors i + j + 2k, i + 2 j + k, i + k respectively and lie in the plane π. (a)
Find (i)
the area of the triangle ABC;
(ii)
the shortest distance from C to the line AB;
(iii)
the cartesian equation of the plane π.
(14)
The line L es through the origin and is normal to the plane π, it intersects π at the point D. (b)
Find (i)
the coordinates of the point D;
(ii)
the distance of π from the origin.
(6) (Total 20 marks)
42
65.
Given any two non-zero vectors a and b, show that
2
a ×b = a
2
2
2
b − (a • b) .
.............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. (Total 6 marks)
66.
Consider the points A(1, −1, 4), B(2, − 2, 5) and O(0, 0, 0). (a)
Calculate the cosine of the angle between OA and AB. (5)
(b)
Find a vector equation of the line L1 which es through A and B. (2)
The line L2 has equation r = 2i + j + 7k + t(2i + j + 3k), where t∈ . (c)
Show that the lines L1 and L2 intersect and find the coordinates of their point of intersection. (7)
(d)
Find the Cartesian equation of the plane which contains both the line L2 and the point A. (6) (Total 20 marks)
67.
⎛ 4 ⎞ ⎜ ⎟ A ray of light coming from the point (−1, 3, 2) is travelling in the direction of vector ⎜ 1 ⎟ ⎜ − 2 ⎟ ⎝ ⎠ 43
and meets the plane π : x + y + 2z − 24 = 0. Find the angle that the ray of light makes with the plane. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. (Total 6 marks)
44
68.
Find the vector equation of the line of intersection of the three planes represented by the following system of equations. 2x − 7y + 5z =1 6x + 3y – z = –1 −14x − 23y +13z = 5 .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. .............................................................................................................................................. (Total 6 marks)
45
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