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Form 2 Mathematics Vectors I Questions and Answers
Given that OA=3i+4j+7k, OB=4i+3j+9k and OC=i+6j+3k, show that points A, B and C are collinear.
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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, #vec ( OA)#=a, #vec (OB)#=b and P lies on AB such that AP: PB=3:5 (a) Find in terms of a and b the vectors (i) #vec (AB)# (ii) #vec (AP)# (iii) #vec (BP)# (iv) #vec (OP)# (b) Point Q is on OP such that #AQ= -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 #OT=QP+ 1/2QR# 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 #PQ=1/3 OR#. 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 + #1/3b# + #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 #triangle# PQR with vertices P (3, 2), Q (5, 0) and R (4, -1). On the same axes, plot P'Q'R', the image of #triangle# 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) #3a + 2b - c + 4 (1/2a-3/2c) + 1/3(6a-9b)# (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) #1/2c-3/2d# (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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