O LEVEL (P1)
VECTORS
QUESTION'S
1
→
1 AB =
–48 , BC→ = 64 .
→
(a) Express AC as a column vector.
→
Answer (a) AC =
[1]
–11
→
(b) It is given that CD =
h .
Find the two possible values of h which will make ABCD a trapezium.
You may use the grid below to help you with your investigation.
A
Answer (b) h = ................... and ............[2]
2
2
C
a
B
2c
4a
O
A
P
→
→
→
In the diagram, OA = 4a, OC = 2c and CB = a.
→
(a) Express BA in terms of a and c.
→
(b) OP = 2a – 4–3 c.
→
→
Explain why OP is parallel to BA .
(c) Find
area of triangle OBA
.
area of triangle OPA
→
Answer (a) BA =............................................................[1]
Answer (b) .................................................................................................................................[1]
3
3
→
→
On the grid in the answer space, OP = p and OQ = q.
→
(a) Given that OR = p – q, mark the point R clearly on the grid.
(b) The point S is shown on the grid.
→
Given that OS = q + hp, find h.
Answer (a)
S
Q
q
O
p
P
[1]
(b) h = ...............................................[1]
4
4
 
3
a = –4
b=
 
–1
7
(a) Express a + 2b as a column vector.
Answer (a)
a + 2b =
(b) (i) Find a .

[1]

Answer (b)(i) a = ..........................[1]
(ii) Given that
b = n , where n is an integer, find the value of n.
a
Answer (b)(ii) n = ............................[1]
5
5
D
C
X
3q
A
2p
B
ABCD is a parallelogram.
X is the point on BC such that BX : XC = 2 : 1.
→
→
AB = 2p and AD = 3q.
Find, in terms of p and q,
→
(a) AC ,
→
Answer (a) AC = ............................ [1]
→
(b) AX ,
→
Answer (b) AX = .......................... [1]
→
(c) XD .
→
Answer (c) XD = ........................... [1]
6
6
3
AB = – 4
(a) Find ⏐AB ⏐.
Answer (a) ...................................... [1]
(b) A is the point (0, 2).
(i) The equation of the line AB may be written 3y + 4x = k.
Find the value of k.
Answer (b)(i) k = ............................ [1]
(ii) Find the coordinates of the midpoint of AB.
Answer (b)(ii) (............ , ........... ) [1]
7
7 In the diagram,
C
B
AB = p, CA = q
p
q
and DC = 3AB .
A
D
(a) Express DA in terms of p and q.
Answer (a) DA = ............................ [1]
(b) E is the point such that BE = kq.
(i) Write down the name given to the special quadrilateral ACBE.
Answer (b)(i) .................................. [1]
(ii) Express AE in terms of p, q and k.
Answer (b)(ii) AE = ....................... [1]
(iii) Given that D, A and E lie on a straight line, find the value of k.
Answer (b)(iii) k = .......................... [1]
8
8
c=
冢32冣
d=
冢–86 冣
(a) Calculate 2c – d .
Answer
冢 冣
Answer
........................................ [1]
[1]
(b) Calculate 兩 d 兩 .
9
9
P
B
C
R
p
Q
A
q
D
In the diagram, ABCD is a parallelogram.
P is the midpoint of BC.
DQ : QP = 1 : 2.
→
→
AB = p and AD = q .
→
(a) Express DP in terms of p and q.
Answer
....................................... [1]
Answer
....................................... [1]
→
(b) Express DQ in terms of p and q.
→
(c) Express AQ in terms of p and q, giving your answer in its simplest form.
Answer
....................................... [1]
Answer
....................................... [1]
→
→
(d) R is the point on BC produced such that BR = k BP .
→
(i) Express AR in terms of p and q and k.
(ii) Given that A, Q and R lie on a straight line, find the value of k.
Answer k = ............................... [1]
10
10
A
p
F
q
E
B
D
C
In the diagram, F is the point on AB where AF = 1 AB.
4
E is the midpoint of AC.
AF = p and AE = q.
(a) Express, in terms of p and q,
(i) FE ,
Answer
........................................ [1]
Answer
........................................ [1]
Answer
........................................ [1]
(ii) BC.
(b) D is the point on BC produced such that BD = kBC.
(i) Express FD in terms of k, p and q.
(ii) Given that F, E and D are collinear, find the value of k.
Answer k = .................................. [2]
11
11 The diagram shows the points
A ( –1, 8 ) and B ( 7, 2 ).
y
A ( –1, 8 )
B ( 7, 2 )
x
O
(a) Find the coordinates of the midpoint of AB.
Answer ( ............... , ...............) [1]
(b) BC =
冢 –34冣
(i) Find the coordinates of C.
Answer ( ............... , ...............) [1]
(ii) Given that 冷 AB + BC 冷 = k , find k.
Answer
k = ............................. [2]
12
12
A
B
2p + q
C
2q – p
D
E
In the diagram, BC = 2p + q, CD = 2q – p and D is the midpoint of CE.
(a) Express, in its simplest form, in terms of p and/or q
(i)
(ii)
CE ,
Answer
................................................ [1]
Answer
................................................ [1]
BE .
(b) Given that AB = kp, express AE in terms of k, p and q.
Answer
................................................ [1]
(c) Given that AE is parallel to BC, find k.
Answer k = ........................................... [1]
13
13 (a)
f
A
J3N
J4N
J-1N
f =K O
g =K O
h =K O
L2P
L1P
L 2P
The vector f and the point A are shown on the grid.
On the grid, mark and label
(i) the point B when AB = f + g,
[1]
(ii) the point C when AC =-2h,
[1]
(iii) the point D when AD = 2f - 3g.
[1]
The rest of this question is on the next page.
14
14
y
A
x
O
A is the point (5, 5)
AB = c
4
m
-3
(a) AB is mapped onto CD by a reflection in the y-axis.
Find CD.
Answer
............................................. [1]
(b) AB is mapped onto AE by a rotation, centre A, through an angle of 90° clockwise.
Find AE .
Answer
............................................. [1]
Answer
............................................. [1]
(c) Find AB .
15
15
C
A
a
O
b
B
D
1
Inthediagram,AisthemidpointofOCandBisthepointonODwhereOB= OD.
3
OA = a andOB = b.
(a) Express,assimplyaspossible,intermsof aandb
(i) AB,
Answer�������������������������������������������� [1]
(ii) CD.
Answer............................................ [1]
(b) EisthepointonCDwhereCE:ED=1:2.
(i) Express BE ,assimplyaspossible,intermsofaand/orb.
Answer�������������������������������������������� [2]
(ii) WhatspecialtypeofquadrilateralisABEC?
Answer�������������������������������������������� [1]
16
16
A
p
Y
Z
C
q
X
B
In the diagram,
1
X is the point on AB where AX = AB ,
4
1
Y is the point on AC where AY = AC ,
3
Z is the point on BC produced where CZ = 2BC.
AY = p and AX = q.
(a) Express, as simply as possible, in terms of p and q,
(i)
XY ,
Answer XY = ............................................. [1]
(ii)
BC ,
Answer BC = ............................................. [1]
(iii)
XZ .
Answer XZ = ............................................. [2]
(b) Hence find XY : YZ.
Answer ................ : ............... [1]
17
17
R
O
a
b
The diagram shows the points O and R and the vectors a and b.
(a) Given that OP = 2a, mark and label the position of P on the grid.
[1]
(b) Given that OQ = 2b – a, mark and label the position of Q on the grid.
[1]
(c) Express OR in terms of a and b.
Answer OR = ............................... [2]
18
18
R
Q
q
S
P
O
p
OPRQ is a parallelogram and S is a point on PR such that PS : SR = 1 : 3.
OP = p and OQ = q.
(a) (i) Express PQ in terms of p and/or q.
Answer ........................................... [1]
(ii) Express QS , as simply as possible, in terms of p and/or q.
Answer ........................................... [1]
(b) T is a point on QS extended such that QT =
4
QS .
3
(i) Express PT , as simply as possible, in terms of p and/or q.
Answer ........................................... [2]
(ii) What can you conclude about the points O, P and T ?
............................................................................................................................................... [1]
19
19
A
C
Q
a
P
O
b
B
OACB is a parallelogram.
OA = a and OB = b.
P and Q are points on OC such that OP = PQ = QC.
(a) Express, as simply as possible, in terms of a and b,
(i)
OP ,
Answer ........................................... [1]
(ii)
BP .
Answer ........................................... [1]
(b) Show that triangles OAQ and CBP are congruent.
[2]