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VECTOR
SPM 1996
1 Given and , find the possible values of p for each of the following cases.
(a) u and v are parallel,(b)
[4 marks]
Answer : (a) ……………………….(b) ……………………….
SPM 19972 Given and , find the value of p if is parallel to the y-axis.
[2 marks]
Answer : p = ……………………..
SPM 1998
3 Given , and R is a point on PQ such that .
Find
(a) ,
(b) .
[5 marks]
Answer : (a) ……………………….(b) ……………………….
SPM 1999
4 Given that ABCD is a parallelogram such that and .
Find
(a) , [2 marks]
(b) the unit vector in the direction of . [3 marks]
Answer : (a) ……………………….(b) ……………………….
SPM 2000
x
y
● 4,8Q● 3,5P
O
5 Given that and such that k is a constant. P is a point on the line AB such
that AP : PB = 1 : 2.
(a) Find in terms of k, a and b. [3 marks]
(b) If the line is extended to Q such that , find the value of k.
[2 marks]
Answer : (a) ……………………….(b) ……………………….
SPM 2002
6 ABCD is a parallelogram. Given that , and such that h and
k are constants.(a) Calculate the value of h and of k. [3 marks]
(b) Hence, calculate the length of the diagonal . [2 marks]
Answer : (a) ……………………….(b) ……………………….
SPM 20037
8
Diagram 2 shows two vectors, .
Express
(a) in the form ,
(b) in the form .
[2 marks]
Answer : (a) ……………………….(b) ……………………….
(a) (b)
DIAGRAM 2
RAJAH 2
DIAGRAM 2
DIAGRAM 2
p = 2a + 3bq = 4a – br = ha + (h – k)b, where h and k are constants.
Use the information above to find the value of h and of k such that r = 3p – 2q. [3 marks]
Answer : h = …...………………….k = …...………………….
9 Diagram 3 shows the parallelogram ABCD such that BED is a straight line.
DIAGRAM 3
Given that , express in terms of p and q :
(a) ,
(b) .
[4 marks]
Answer : (a) = ..……………….
(b) = ..……………….
SPM 200410 Given , and ,find in terms of the unit vectors, and ,
(a) ,
(b) the unit vector in the direction of .
[4 marks]
Answer : (a) ……………………….(b) ……………………….
11 Given , and , find the value of m and of p such that
. [4 marks]
x
y
Q
P
O
R
Answer : m = …………………….. p = …………………......
SPM 2005
12 Diagram 4 shows vector drawn on a Cartesian plane.
(a) Express in the form .
(b) Find the unit vector in the direction of
[2 marks]
Answer : (a) = ...………………
(b) ……....……………….
13
SPM 200614
5 B
O
y
x
A (4, 3)
Diagram 6 shows two vectors, .
Express
(a) in the form ,
(b) in the form .
[3 marks]
Answer : (a) =…………………...
(b) = …………………...
●
y
xO
2
4
6
2 4 6 8 10 12
A
DIAGRAM 5
DIAGRAM 4
Diagram 5 shows the parallelogram OPQR,
drawn on a Cartesian plane. Given
and .
Find [3
marks]
Answer : = ….……………….
DIAGRAM 5
DIAGRAM 5
-10i + j
15 The points P, Q and R are collinear. It is given that and ,
where k is a constant.Find(a) the value of k,(b) the ratio of PQ : QR.
[4 marks]
Answer : (a) k = ………………………(b) …………………………..
SPM 200716
17 The following information refers to the vectors .
Find
DIAGRAM 6
●
O A
BC
Dy5
x9
DIAGRAM 3
Diagram 3 shows a rectangle OABC and the point D lies on the straight line OB.
It is given that OD = 3DB. Express , in terms of
and . [3 marks]
Answer : = ………………….
O
B
A
● X
a7
b7
(a) the vector ,(b) the unit vector in the direction of .
[4 marks]
Answer : (a) ....…………………......(b) .......…………………...
PER PERAK 2004
18 Given that PQRS is a parallelogram where . Find the unit
vector in the direction of . [4 marks]
Answer : ….……………………….
19 Given If is parallel to , find the value of m. [2 marks]
Answer : ….……………………….
20
PER PERAK 200621
Diagram 3 shows the triangle OAB, such that
, and BX : XA = 3 : 4.
Find the vector in terms of
[4 marks]
Answer : ….……………………….
DIAGRAM 3
x
y
●
●
O
DIAGRAM 3
Diagram 5 shows two vectors and drawn
on a Cartesian plane.Find in terms of and ,
(a)
(b) the vector unit in the direction
[4 marks]
Answer : (a) ……………………….(b) ……………………….
22 Given and , find the value of t if p is parallel to q. [4 marks]
Answer : t = ..……………………..
PER PERAK 2007
23 The points O(0, 0), A(3, 12) and B(0, 4) are three points on a Cartesian plane. Given ,
and M is a point such that . Find
(a) the coordinates of M,
(b) .
[4 marks]
Answer : (a) ……………………….(b) ……………………….
24
PER MELAKA 2004
25 The vector has a magnitude of 6 units. Express p in terms of q. [2 marks]
Answer : p = ..………………………
P Q
RS
● M
DIAGRAM 5
(a) (6, 16) (b) √73 = 8.544
Diagram 3 shows parallelogram PQRS with
straight line PMR. Given that ,
and PM = 2MR.
Express in terms of x and y.
[3 marks]
Answer : .…………………………..DIAGRAM 3
5i + 5j
3
26 The vector has a magnitude of 50 units and is parallel to the vector . Find
(a) the unit vector in the direction of ,
(b) in form of the column vector .
[3 marks]
Answer : (a) ……………………….(b) ……………………….
PER MELAKA 200527 Given that a = 10j and b = . Find the unit vector that is parallel to a – 3b. [4 marks]
Answer : ….………………………..
28 Given that and . If p is parallel to q, find all the possible values of m. [4 marks]
Answer : ….……………………….
PER MELAKA 200629
30 Given and are two parallel vectors, find the value of a. [2 marks]
Answer : a = ..……………………..
PER MELAKA 2007
P(2, 2)●
● 5,2Q
Ox
y
DIAGRAM 4
Diagram 4 shows the vector drawn on a
Cartesian plane.
(a) Express in terms of and .
(b) Find .
[2 marks]
Answer : (a) ……………………….(b) ……………………….
4
(a) −4i+3j (b) 5
3 , −2
31 Given that , and C is a point that lies on the straight line AB such that
5AC = 2AB. Find
(a)
(b) .
[3 marks]
Answer : (a) ……………………….(b) ……………………….
32
PER JOHOR 2005
33 ABCD is a rhombus where . Find
(a) ,
(b) the unit vector in the direction of .
[4 marks]
Answer : (a) ……………………….(b) ……………………….
34 Given that the vectors and . Find the value of h such that
(a) a is parallel to b,(b) a and b have equal magnitude.
[4 marks]
Answer : (a) ……………………….(b) ……………………….
ab
O●
DIAGRAM 4
Diagram 4 shows vectors a and b drawn on square grids. On the diagram, draw and label the direction of
the line which represents the vector .
[3 marks]
(a) −2i – 4j (b)
PER JOHOR 200635 Given and , find . [2 marks]
Answer : ….……………………….
36 Diagram 3 shows the vectors and . On the diagram, draw the vector such
that . [2 marks]
PER JOHOR 200737
OA
B
DIAGRAM 3
A B
CD
E
DIAGRAM 3
In Diagram 3, ABCD is a quadrilateral. The straight lines AC and BD intersect at point E. Given that
, , DC : AB = 3 : 4
and AE : EC = 3 : 2.Express in terms of u and v:
(a)
(b)
[4 marks]
Answer : (a) ……………………….(b) ……………………….
√5
38 Given that vectors and , find the value of k in each of the following cases:
(a) .
(b) u and v are parallel. [3 marks]
Answer : (a) ……………………….(b) ……………………….
PER TERENGGANU 200439 Given a = 2i + j and b = 3i + mj, find all the possible values of m if . [4 marks]
Answer : m = ..…………………….
PER TERENGGANU 2005
40 Given that O is the origin, the point and .
(a) State in terms of i and j.
(b) Find .
[3 marks]
Answer : (a) ……………………….(b) ……………………….
41 If and , find the value of p for each of the following cases:(a) a + b is parallel to the x-axis,(b) a is parallel to b.
[4 marks]
Answer : (a) p = .…………………..(b) p = …………………...
PER TERENGGANU 2006
(a) 3i – 4j (b) √74
(a) −7 (b) −5
(a) 1 (b) 9
42 Given that the unit vector in the direction is and units. Find
(a)
(b) the coordinates of A.[3 marks]
Answer : (a) ……………………….(b) ……………………….
43 Given that and . Find the value of m for each of the following cases:
(a) u and v are parallel,(b) 2u + v is parallel to the y-axis.
[4 marks]
Answer : (a) m = …………………..(b) m = …………………..
PER TERENGGANU 200744 Given and , find the values of k if . [2 marks]
Answer : k = ………………………
45 Given the points
(a) Express in terms of and .
(b) Find the coordinates of R if .
[4 marks]
Answer : (a) = ..……………….
(b)………..………………PER NS 200446 Diagram 4 shows the vectors s and t, and the unit vectors a and b.
ts
ba
(a) 8i – 6j (b) (8, −6)
(a) 4 (b) −3
R
P
Q
S ●
Given , express r in terms of a and b. [3 marks]
Answer : r = ..…………………….
47 Given and , find the value of p if . [3 marks]
Answer : p = ..…………………….
PER NS 200548
49
PER NS 200650
●
y
Ox
p
DIAGRAM 2
DIAGRAM 3
Diagram 2 shows the triangle PQR such that = 2x
and Given PS : PR = 2 : 3.
Find . [3 marks]
Answer : ….……………………….
Diagram 3 shows the position vector P on a Cartesian plane. Given .Find the value of m. [3 marks]
Answer : ….……………………….
DIAGRAM 4
Diagram 4 shows the vectors and drawn on
square grids. Express in terms of a and b
(a)
(b)
[4 marks]
Answer : (a) ……………………….(b) ……………………….
B
A ●
●
C
D
O
CD
A B
E
51 Given the points A(3, 0), B(7, 8) and C(1, k).
(a) Express vector in terms of i and j.
(b) Find the value of k if the vectors and are parallel.
[4 marks]
Answer : (a) ……………………….(b) ……………………….
PER NS 2007
52 Given that and . If is parallel to , find the possible values of a.
[3 marks]
Answer : .…………………………..
53 Given that , and , where p and q are constants. Find the value of p and q if . [3 marks]
Answer : p = ……………………... q = ……………………...
PER SELANGOR 200454
Diagram 4 shows a parallelogram ABCD. Given
, find
[2 marks]
Answer : ….……………………….
DIAGRAM 4
(a) 4i + 8j (b) 2
(a) b – a (b) 2b
●
●
x
y
O
55
56 Given that . Find all the values of m if . [3 marks]
Answer : ….……………………….
PER SELANGOR 2005
57 Given that and . Find in terms of the unit vectors i and j,
(a) ,(b) the unit vector in the direction of .
[3 marks]
Answer : (a) ……………………….(b) ……………………….
58 Given that the points . Find the value of p if
(a) is parallel to
(b) the points A, B and C are collinear.[4 marks]
Answer : (a) ……………………….(b) ……………………….
PER SELANGOR 200659
DIAGRAM 4
In Diagram 4, and .
(a) Express 2a + b in the form of column vector.(b) Find .
[4 marks]
Answer : (a) ……………………….(b) ……………………….
DIAGRAM 4
O
A
B
a b
P
QDiagram 4 shows the vectors and
drawn on a Cartesian plane.
Given that , express
.
[2 marks]
Answer : ….……………………….
(a) ⅔ (b) −11
60 Given that . Find the value of n such that A, B and C are
collinear. [2 marks]
Answer : n = ..……………………..
PER SELANGOR 200761
DIAGRAM 4
DIAGRAM 5
x
y
●
●
O
3a – 2b
Diagram 5 shows two vectors and drawn
on a Cartesian plane.
(a) Express in the form .
(b) Given that , find the value of
m and of n. [3 marks]
Answer : (a) ……………………….(b) m = ….……………….
n = …………………..
A●● C
●B
x
y
O
62
PER KELANTAN 200563
C
A
a 2 4
y
4
2
2
4
b
B
xO
Diagram 4 shows vectors
(a) Express the vector in terms of
.
(b) Write the vector in the form .
[4 marks]
Answer : (a) ……………………….(b) ……………………….
Diagram 6 shows a triangle OAB drawn on a Cartesian plane. The point C is the midpoint of AB.
Given that and .
(a) Express in the form of .
(b) Given that , find the value
of h.
[4 marks]
Answer : (a) ……………………….(b) ……………………….
DIAGRAM 6
PER KELANTAN 200664
65 Given , find the unit vector in the direction of [3 marks]
Answer : ….……………………….
PER KELANTAN 2007
66 Given that and . Find
(a)
(b) the unit vector in the direction of
[3 marks]
Answer : (a) ……………………….(b) ……………………….
DIAGRAM 4
2,1Q
R(k, m)
2,3 P
x
y
O
Diagram 6 shows the triangle PQR. Given
, find the value of k and of m.
[4 marks]
Answer : k = ……………………… m = ……………………...
DIAGRAM 6
k = 2 , m = 1
67 Given that , and . Find the value of m and of n if
[3 marks]
Answer : m = ……………………...
PER KEDAH 2005
68 Diagram 3 shows the vectors , , drawn on square grids.
Express in terms of r and/or s.
(a)
(b)
[4 marks]
Answer : (a) ……………………….(b) ……………………….
69 Given the vector
(a) Find the unit vector in the direction of .
(b) If ABCD is a parallelogram, find the vector .
[3 marks]
Answer : (a) ……………………….(b) ……………………….
PER KEDAH 2006
sS
O
r
R
P
Q
DIAGRAM 3
m = 2 , n = 8
70 Given where O is the origin, find the unit vector in the direction of
[2 marks]
Answer : ….……………………….
71
PER KEDAH 200772 Given that and . Find the values of m if a is parallel to b. [2 marks]
Answer : m = ………………………
73 Given , find the unit vector in the direction of q. [2 marks]
Answer : .…………………………..
PER SMK SULTAN ABU BAKAR 200774
●
A
B
CP
DIAGRAM 3
Diagram 3 shows the vectors and .
P is a point on CB such that CP : PB = 1 : 2.
Find in terms of i and j. [2 marks]
Answer : ….……………………….
●C(2, 7)
● B(5, 3)
● A(0, h)O
x
y
DIAGRAM 3
−3
Diagram 3 shows vectors and .
(a) Express in the form of .
(b) Given that , find the value
of h. [3 marks]
Answer : (a) ……………………….(b) h = ….………………..
(a) 3i – 4j (b) −4
75
PER MRSM 2004
76 Diagram 5 shows the vectors , drawn on square grids.
A
B
C
D
E
DIAGRAM 4
O
P
Q
q
p
R
S
Diagram 4 shows parallelogram ABCD with a straight line BED.
Given that , and 4DE = EB.
Express in terms of p and q:
(a) ,
(b) .
[4 marks]
Answer : (a) = ..……………….
(b) ……..……………
Given and , express in terms of p and/or q
(a) ,
(b) .
[3 marks]Answer : (a) ……………………….
(b) ……………………….
PER MRSM 200577 Given and , find . [2 marks]
Answer : ….……………………….
PER MRSM 200678 Given the vectors and are parallel, express p in terms of q. [3 marks]
Answer : ….……………………….
79
PER MRSM 200780
DIAGRAM 5
O P
QR ●Y
a
b
DIAGRAM 3
Diagram 3 shows the parallelogram ABCD where
and .
Given Y is the midpoint of QR, express in
terms of a and b. [3 marks]
Answer : ….……………………….
y
xO
A
B
Diagram 3 shows triangle OAB, with origin O. The coordinates of points A and B are and respectively.Find
(a) ,
(b) the possible unit vectors that are parallel to .
[4 marks]
Answer : (a) ……………………….(b) ……………………….
81 The vectors and are non-parallel. Given that , where m and n are constants. Find the value of m and n. [2 marks]
Answer : m = ……………………... n = ……………………...
PER SBP 200482 Given that the points and O is the origin.
(a) Express in the form of .
(b) Find the value of .
[3 marks]
Answer : (a) ……………………….(b) ……………………….
83 The information below shows the vectors and
Find the value of h and of k. [3 marks]
Answer : h = ……………………….k = ……………………….
PER SBP 2005
84 Given the points and . Point C is the midpoint of AB such that .
DIAGRAM 4
, where h and k are constants
DIAGRAM 3
Find (a)
(b)
[4 marks]
Answer : (a) ……………………….(b) ……………………….
PER SBP 2006
85 In Diagram 4, and represent the column vectors and respectively.
86 Given and , find the value of k if and are parallel vectors.[2 marks]
Answer : k = ..…………………….
PPT T5 SBP 200787
A
B
C
DIAGRAM 5
(a) Express as a column vector.
(b) On Diagram 5, draw the vector
such that .
[3 marks]
Answer : (a) ……………………….(b) .....................................
B O
A
P
DIAGRAM 3
Diagram 3 shows a triangle OAB such that
= a , = b and .
Find in terms of a and b.
[2 marks]
Answer : ….……………………….
88
PER SBP 200789
90 Given and , find . [2 marks]
Answer : ….……………………….
S(3, 1)
T(8, 5)
Ox
y
O
2,3 Q
y
x
DIAGRAM 4
Diagram 4 shows vectors , and
drawn on a Cartesian plane.Find the value of m and n such that
[4 marks]
Answer : m = ……………………n = ……………………
DIAGRAM 4
Diagram 4 shows the points P and .
Find (a) in terms of and ,
(b) the unit vector in the direction of
.
[3 marks]
Answer : (a) = ..……………….
(b)………..………………
10
