1
1
B
NOT TO
SCALE
b K
L
O a A
The diagram shows a triangle OAB and a parallelogram OALK.
The position vector of A is a and the position vector of B is b.
K is a point on AB so that AK : KB = 1 : 2.
Find the position vector of L, in terms of a and b.
Give your answer in its simplest form.
................................................... [4]
[Total: 4]
2
, and .
Calculate the ratio AD : DB.
.............................. : .............................. [2]
[Total: 2]
Find .
................................................... [2]
[Total: 2]
3
4
R Q
NOT TO
a SCALE
O b P
The diagram shows a trapezium OPQR.
O is the origin, and .
(a) Find in terms of a and b in its simplest form.
................................................... [2]
(b) When PQ and OR are extended, they intersect at W.
Find the position vector of W.
................................................... [2]
[Total: 4]
4
5 The magnitude of the vector is 29.
Find the value of k.
k = ................................................... [3]
[Total: 3]
6
P
NOT TO
S SCALE
a
O b Q
S is a point on PQ such that PS : SQ = 4 : 5.
Find , in terms of a and b, in its simplest form.
................................................... [2]
[Total: 2]
5
OABC is a parallelogram.
and .
E is the point on AB such that AE : EB = 3 : 1.
Find , in terms of p and q, in its simplest form.
................................................... [2]
[Total: 2]
Find .
................................................... [2]
[Total: 2]
6
O is the origin and OPQR is a parallelogram.
SOP is a straight line with SO = OP.
TRQ is a straight line with TR = RQ.
STV is a straight line and ST : TV = 2 : 1.
and .
(a) Find, in terms of a and b, in its simplest form,
(i) the position vector of T,
................................................... [2]
(ii) .
= ................................................... [1]
(b) Show that PT is parallel to RV.
[2]
[Total: 5]
7
10
(a) Find .
[2]
(b) Find .
................................................... [2]
[Total: 4]
8
11
In the diagram, O is the origin, OT = 2TD and M is the midpoint of TC.
and .
Find the position vector of M.
Give your answer in terms of c and d in its simplest form.
................................................... [3]
[Total: 3]
12
O is the origin, , and .
and .
Find, in terms of p and q, in its simplest form
9
(a) ,
................................................... [2]
(b) the position vector of M.
................................................... [2]
[Total: 4]
13
OABC is a parallelogram and O is the origin.
CK = 2KB and AL = LB.
M is the midpoint of KL.
and .
Find, in terms of p and q, giving your answer in its simplest form
10
(a) ,
................................................... [2]
(b) the position vector of M.
................................................... [2]
[Total: 4]
14
ABCD is a parallelogram.
N is the point on BD such that BN : ND = 4 :1.
and .
Find, in terms of s and t, an expression in its simplest form for
(a) ,
= ................................................... [1]
11
(b) .
= ................................................... [3]
[Total: 4]
15
ABCD is a parallelogram with and .
ABM is a straight line with AB : BM = 1 : 1.
ADN is a straight line with AD : DN = 3 : 2.
(a) Write , in terms of p and q, in its simplest form.
= ................................................... [2]
12
(b) The straight line NM cuts BC at X.
X is the midpoint of MN.
Find the value of k.
k = ................................................... [2]
[Total: 4]
16 O is the origin, and .
Find the position vector of B, in terms of x and y, in its simplest form.
................................................... [2]
[Total: 2]
13
17
In the diagram, OABC is a parallelogram.
OP and CA intersect at X and CP : PB = 2 : 1.
= a and = c.
(a) Find , in terms of a and c, in its simplest form.
= ................................................... [2]
(b) CX : XA = 2 : 3
(i) Find , in terms of a and c, in its simplest form.
= ................................................... [2]
(ii) Find OX : XP.
OX : XP = .................... : .................... [2]
[Total: 6]
14
18
OAB is a triangle and C is the mid-point of OB.
D is on AB such that AD : DB = 3 : 5.
OAE is a straight line such that OA : AE = 2 : 3.
= a and = c.
(a) Find, in terms of a and c, in its simplest form,
(i) ,
= ................................................... [1]
(ii) ,
= ................................................... [1]
(iii) ,
= ................................................... [1]
(iv) .
= ................................................... [2]
15
(b)
Find the value of k.
k = ................................................... [1]
[Total: 6]
19
(a) Find 3a – 2b.
[2]
(b) Find .
................................................... [2]
16
(c)
Write down two simultaneous equations and solve them to find the value of m and the value of n.
Show all your working.
m = ...................................................
n = ................................................... [5]
[Total: 9]
17
20
O is the origin, and .
QT : TP = 2 : 1
Find the position vector of T.
Give your answer in terms of p and q, in its simplest form.
................................................... [2]
[Total: 2]
21 In the diagram, O is the origin, and .
18
(a) Find , in terms of a and b, in its simplest form.
................................................... [2]
(b)
Find the position vector of E, in terms of a and b, in its simplest form.
................................................... [2]
[Total: 4]
19
PQRS is a parallelogram with diagonals PR and SQ intersecting at X.
and .
Find in terms of a and b.
Give your answer in its simplest form.
22 = ................................................... [2]
[Total: 2]
23
Find
(a) ,
= ...................................................
[3]
20
(b) .
= [2]
[Total: 5]
The diagram shows a regular hexagon ABCDEF.
and .
24 Find , in terms of p and q, giving your answer in its simplest form.
= ................................................... [2]
[Total: 2]
21
25 Find the magnitude of the vector .
................................................... [2]
[Total: 2]
26
Find .
Answer ................................................... [2]
[Total: 2]
27
B
NOT TO
SCALE
b
C
O a A
In the diagram, O is the origin, = a and = b.
C is on the line AB so that AC : CB = 1 : 2.
Find, in terms of a and b, in its simplest form,
22
(a) ,
Answer(a) = ...................................................
[2]
(b) the position vector of C.
Answer(b) ................................................... [2]
[Total: 4]
