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Form 2 Mathematics Vectors I Questions and Answers
If
1
3
a-
1
5
b = 0, write a as a multiple of b.
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1.
In the figure beside, AB=p, AC=q, AD=
3
5
AC and CE=
2
3
CB. Express DE in terms of p and q.
2.
In a triangle ABC, D is the midpoint of AB and E is a point on BC such that BE=
2
3
BC. If AD=p and AC=q, express EC in terms of p and q.
3.
A point T divides a line AB internally in the ratio 5:2. Given that A is (-4, 10) and B is (10, 3), find the coordinates of T.
4.
OABC is a trapezium such that the coordinates of O, A, B and C are (0,0), (2,-1), (4,3) and (0,y). (a) Find the value of y (b) M is a midpoint if AB and N is a midpoint of OM. Show that A,N and C are collinear.
5.
OABC is a trapezium in which OA=a, OC=c and CB=3a. CB is produced to D such that CB:BD=3:1. E is a point on AB such that AB=2AE. Show that O, E and D are collinear.
6.
In the figure below CA=b, CB=a, AX=XY and AY=YB. Express CX in terms of a and b.
7.
In the figure below OC=3CA and OD =3DB. By taking OA=a, OB=b show that CD//AB.
8.
In the figure below, ABCD is a parallelogram. AOC and BOD are diagonals of the parallelogram. Show that the diagonals of the parallelogram bisect each other. Give reasons.
9.
The figure below is a right pyramid with a rectangular base ABCD and VO as the height. The vectors AD=a, AB=b and DV=c (a) Express (i) AV in terms of A and c (ii) BV in terms of a, b and c (b) M is a point on OV such that OM: MV=3:4. Express BM in terms of a, b and c. Simplify your answer as far as possible.
10.
In the figure below, OA=3i+4j and OB=8i-j. C is a point on AB such that AC:CB=3:2, and D is a point such that OB//CD and 2OB=CD Determine the vector DA in terms of i and j
11.
ABC is triangle and P is a point on AB such that p divides AB internally in the ratio 4:3. Q is a point on AC such that PQ is parallel to BC. If AC=14cm, (i) State the ratio AQ:QC (ii) Calculate the length of QC
12.
In the figure below, KLMN is a trapezium in which KL is parallel to NM and KL=3NM Given that KN=w, NM=u and ML=v, show that 2u=v+w
13.
In triangle OAB,
→
O
A
=a,
→
O
B
=b and P lies on AB such that AP: PB=3:5 (a) Find in terms of a and b the vectors (i)
→
A
B
(ii)
→
A
P
(iii)
→
B
P
(iv)
→
O
P
(b) Point Q is on OP such that
A
Q
=
-
5
8
a
+
9
40
b
. Find the ratio OQ:QP
14.
The coordinates of points O, P, Q and R are (0, 0), (3, 4), (11, 6) and (8, 2) respectively. A point T is such that vectors OT, QP and QR satisfy the vector equation
O
T
=
Q
P
+
1
2
Q
R
Find the coordinates of T.
15.
(a) If A, B and C are the points (2, -4), (4,0) and (1,6) respectively, use the vector method to find the coordinates of the points D given that ABCD is a parallelogram. (b) The position vectors of points P and Q are p and q respectively. R is another point with position vector,
r
=
3
2
q
-
1
2
p
. Express in terms of p and q (i) PR (ii) RQ, hence show that P, Q and R are collinear.
16.
The points P,Q, R and S have position vectors 2p, 3p,r and 3r respectively, relative to an origin O. A point T divides PS internally in the ratio 1:6. (a) Find in the simplest form the vectors OT and QT in terms of p and r (b) (i) Show that the point Q,T and R lie on a straight line (ii) Determine the ratio in which T divides QR
17.
Two points P and Q have coordinates (-2, 3) and (1, 3) respectively. A translation maps point P to P’ (10, 10). (a) Find the coordinates of Q’, the image of Q under the translation. (b) The position vectors of P and Q in (a) above are p and q respectively. Given that mp-nq=
(
-
12
9
)
, find the values of m and n
18.
In the diagram below, the coordinates of points A and B are (1, 6) and (15, 6) respectively. Point N is on OB such that 3ON=2OB. Line OA is produced to L such that OL= 3OA. (a) Find vector LN (b) Given that a point M is on LN such that LM:MN=3:4, find the coordinates if M . (c) If line OM is produced to T such that OM:MT=6:1 (i) Find the position vectors of T (ii) Show that points L, T and B
19.
The position vectors of points A and B with respect to the origin O are
(
-
8
5
)
and
(
12
-
5
)
respectively. Point M is the midpoint of AB and N is the midpoint of OA. (a) Find: (i) the coordinates of N and M (ii) the magnitude of NM (b) Express vector NM in terms of OB (c) Point P maps onto P’ by a translation
(
-
5
8
)
. Given that OP=OM+2MN, find the coordinates of P’.
20.
Vector OA=
(
2
1
)
and OB=
(
6
-
3
)
. Point C is on OB such that CB=2OC and point D is on AB such that AD =3DB. Express CD as a column vector.
21.
In the figure below, OPQR is a trapezium in which PQ is parallel to OR and M is the midpoint of QR, OP=p, OR=r and
P
Q
=
1
3
O
R
. Find OM in terms of p and r
22.
Vectors OP=6i+j and OQ=-2i+5j. A point N divides PQ internally in the ratio 3:1. Find PN in terms of i and j
23.
Given that OA=2i+3j and OB= 3i -2j. Find the magnitude of AB to one decimal place.
24.
Given that p=5a-2b where a=
(
3
2
)
and b=
(
4
1
)
. Find (a) Column vector p; (b) P’, the image of P under translation vector
(
-
6
4
)
.
25.
The position vectors of point P, Q and R are OP=
(
-
3
6
)
, OQ=
(
2
1
)
, OR=
(
4
-
1
)
. Show that P, Q and R are collinear.
26.
Given that a=
(
4
3
)
, c=
(
-
2
-
5
)
and 3a-2b+4c=
(
10
-
19
)
, find b
27.
Given that OA=3i+4j+7k, OB=4i+3j+9k and OC=i+6j+3k, show that points A, B and C are collinear.
28.
Given that OA=
(
2
3
)
and OB=
(
-
4
5
)
. Find the midpoint M of AB
29.
Given that b=
(
2
4
)
, c=
(
3
2
)
and a=3c-2b, find the magnitude of a, correct to 2 decimal places.
30.
Given that t = 2a - b and r = b + 3a, find, in terms of a and b, the following vectors: (a) 2t - 2r (b) –t +
1
2
r (c) 2r – t
31.
Simplify: (a) 3(a – 2b) – 4(5a – b) (b)
1
3
(a - b) +
1
4
(3b - 4a) (c) 3a – 2b – c + 4(
1
2
a-
3
2
c) +
1
3
(6a –9b)
32.
If
1
3
a-
1
5
b = 0, write a as a multiple of b.
33.
If a =
(
3
1
)
, b =
(
-
2
7
)
, c =
(
6
5
)
and d=
(
0
-
5
)
, find: (a) 3a - 2b (b)
1
3
a +
1
3
b
+
1
3
c (c) 10a+15b+23d (d) The scalars r and s such that ra + sc = 9b
34.
If x =
(
-
1
0
)
and z=
(
3
-
1
)
, find –y, given that x + y = z.
35.
Find the co-ordinates of P if OP = OA + OB – OC and the co-ordinates of points A, B, C are (3, 4), (-3, 4) and (-3, -4) respectively.
36.
In the figure below, OAB is a triangle. A is the point (2, 8) and B the point (10, 2). C, D, and E are midpoints of OA, OB and AB respectively. (a) Find the co-ordinates of C and D. (b) Find the length of the vectors CD and AB.
37.
Find the co-ordinates of the midpoint of AB in each of the following cases: (a) A (-4, 3), B (2, 0) (b) A (0, 0), B (a, b)
38.
Triangle OAB is such that OA = a and OB = b. C lies on OB such that OC: CB = 1:1. D lies on AB such that AD: DB=1:1 and E lies on OA such that OA: AE = 3:1. Find: (a) OC (b) OD (c) OE (d) CD (e) DE
39.
The co-ordinates of A, B, C and D are (-2, 5), (1, 3), (3, -2) and (2, -4) respectively. If A' is (-5, 6) under a translation T, find the co-ordinates of B', C' and D' under T.
40.
A point P (-2, 3) is given the translation
(
-
1
4
)
. Find the point P', the image of P, under the translation. If P' is given a translation
(
1
-
2
)
, find the co-ordinates of P", the image of P'. Find a single translation that maps P onto P".
41.
The point A (3, 2) maps onto A' (7, 1) under a translation T_1. Find
T
1
. If A' is mapped onto A" under translation
T
2
given by
(
-
3
5
)
, find the co-ordinates of A". Given that
T
3
(
A
)
= A", find
T
3
.
42.
The image of point (6, 4) is (3, 4) under a translation. Find the translation vector.
43.
Draw
△
PQR with vertices P (3, 2), Q (5, 0) and R (4, -1). On the same axes, plot P'Q'R', the image of
△
PQR under a translation given by
(
-
4
1
)
.
44.
The figure below shows ABC a triangle in which the midpoints of AB, BC and AC are E, F and D respectively. Vector AB = -2b while BC = 2a. Rewrite each of the following vectors in terms of a and b: (a) BF (b) AF (c) AC (d) DC (e) DA (f) BD
45.
Given that t = 2a - b and r = b + 3a, find, in terms of a and b, the following vectors: (a) 2t (b)
1
2
r (c) -t (d) -2r
46.
The figure below shows a regular hexagon PQRSTU. PQ = a, QR = b and RS = c. Write in terms of a, b and c each of the following: (a) QP (b) PS (c) TQ (d) TP (e) RT (f) UQ (g) US
47.
Simplify: a)
3
a
+
2
b
-
c
+
4
(
1
2
a
-
3
2
c
)
+
1
3
(
6
a
-
9
b
)
(b) 2u - 3v + 2(w - u) + 3(u + v) (c) (p - q) + (r - p) + (q - r)
48.
If
a
=
(
3
1
)
,
b
=
(
-
2
7
)
,
c
=
(
6
5
)
and
d
=
(
0
-
5
)
, find: (a) 5b-3a (b)
1
2
c
-
3
2
d
(c) 3b+c
49.
Find scalars m and n such that
m
(
4
3
)
+
n
(
-
2
1
)
=
(
0
-
3
)
50.
Find the co-ordinates of the midpoint of AB in each of the following cases: (a) A (0, 4), B (0, -2) (b) A (-3, 2), B (4, 2)
51.
For each of the points below, give the image when it is translated by the given vector: (a)
(
2
,
3
)
;
(
4
3
)
(b)
(
0
,
0
)
;
(
-
2
1
)
(c)
(
-
2
,
-
4
)
;
(
4
3
)
52.
For each of the points below, give the image when it is translated by the given vector: (a)
(
-
4
,
3
)
;
(
4
-
3
)
(b)
(
7
,
0
)
;
(
-
8
-
1
)
(c)
(
8
,
-
2
)
;
(
-
10
4
)
53.
Below are images of points which have been translated by the vectors indicated; Find the corresponding object points: (a)
(
2
,
3
)
;
(
1
1
)
(b)
(
0
,
0
)
;
(
4
3
)
(c)
(
9
,
17
)
;
(
2
3
)
54.
Below are images of points which have been translated by the vectors indicated; Find the corresponding object points: (a)
(
5
,
6
)
;
(
-
2
-
4
)
(b)
(
7
,
-
2
)
;
(
-
5
2
)
(c)
(
-
2
,
4
)
;
(
7
5
)
55.
Below are images of points which have been translated by the vectors indicated; Find the corresponding object points: (a)
(
3
,
3
)
;
(
-
4
-
10
)
(b)
(
-
5
,
-
11
)
;
(
-
12
-
0
)
(c)
(
4
,
3
)
;
(
0
0
)
(d)
(
a
,
b
)
;
(
c
d
)
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