VECTOR

A Vector is a quantity that has a magnitude and a direction.

Examples: Displacement
Velocity
Acceleration
Force
Momentum

A Scalar is a quantity that has a magnitude only

Examples: Distance
Speed
Area
Volume
Mass

a

A

The Vector from A to B can be denoted by

capital letters as AB or AB or small letter a


The magnitude of a vector or is denoted by and is read as modulus

of vector .

Zero Vector is a vector whose magnitude is zero and is denoted by

Negative Vector is a vector that has the same magnitude but is of the opposite
direction from the reference vector.

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Two vector are equal if both of them are of the same magnitude and direction.

Q
b

B

P
a

AB = PQ
A

MULTIPLICATION OF A VECTOR BY A SCALAR

The multiplication of a vector by a scalar k gives a vector k .

MEANING:
The magnitude of k = k times the magnitude of , that is .

The direction of k is the same as if k is positive

-2 
b

2b


b


2
Example:

1. The vector has a magnitude of 25 km h due east. If , state the magnitude

and the direction of the vector .

2. If vector is a velocity of 20 ms due north and is a velocity of 20 ms due

south, is ?

3. If a represents displacement of 8 km due east and reperesents a displacement of

8 km due west, state the relation between and .

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PARALLEL VECTOR is one of the vectors can be expressed in terms of the other
vector. In other words, if one vector is the scalar multiple of another vector, both
vecors are parallel.

If AB = kPQ or PQ = hAB , where h and k are

constant, line AB is parallel to line CD
A
P

If the vector and are not parallel and , then h = 0 and k = 0.

COLLINEAR CONDITION OF THREE POINTS

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Example collinear
1.
R

In the above diagram, OPQR is a straight line such that If ,

express each of the following vectors in terms of .

2. If vector and are not parallel and (k – 5) = (3 + t) , find the value of k and or
t.

3. and are two vectors that are not parallel to each other. If (3x-2) = (2y + 7) ,
Find the value of x and y

5
4. The diagram, ABCDEF is a regular hexagon such that
Express each of the following vectors in terms of

A B

F C

E D

6
ADDITION OF VECTORS
1. Parallel vectors - - can be added algebraically and its direction is maintained.

2. Triangle law of Addition.

Connecting the starting point with endpoint.

3. Parallelogram law of Addition.

4. Polygon Law of Addition.

E C

A B

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Sharpen your skills
1.
S

Base on the above figure , find the resultant vector for each of the following.

2. Given two vectors below

Construct and label clearly each of the following vectors

3. In the diagram below, PQRS is a parallelogram. It is given and .

PR and QS intersect each other at point M. Find each of the following vectors in
terms of .

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 S R

2 Q

4. In the bottom figure, E is a point on the line BD such that DE = DB. The line AB

is parallel to the line DC, DC = AB ,

D
C

A B

a) Express each of the following vectors in terms of .

i. ii. iii.
b) Hence, prove that BC is parallel to AE.

9
5. In the diagram below, N is the midpoint of the line RT,

N
R

P Q

a) Express each of the following vectors in terms of

i. ii.

b) Determine whether the points P, N and S are collinear.

S is a common point, so …..

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6. In the below diagram, OA is produced to C such that
OC = 3 OA. OB is produced to D such that OD = 2 OB. The lines AD and BC
intersect at P.
C

P
O

a) Find each of the following vectors in terms of .

i. ii.

b) Given that , express in terms of

i. m, ii. n,

C) Hence, find the value of m and of n.

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Vector in the Cartesian Plane

A unit vector is a vector with magnitude 1 unit.

B 4

D
A

-10 -5 5 10

C E

-2

-4

-6

-8

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PAPER 1 (2005)

15 Diagram 1 shows vector drawn on a Cartesian Plane.

6
4 A

O 2 4 6 8 10 12 x
Diagram 1

a) Express in the form .

b) Find the unit vector in the direction of .

2 Marks

16 Diagram 2 shows a parallelogram OPQR drawn on a Cartesian plane.

y
Q

R
P

O
x
Diagram 2

13
It is given that
Find
3
Marks

PAPER 2 (2005)

6. In Diagram 3, ABCD is a quadrilateral. AED and EFC are straight lines.

E F C

A B

Diagram 3

It is given that

a) Express in terms of
i) ii) .

b) Show that the points B, F and D are collinear.

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c) If and , find .

8 Marks

SPM 2006, PAPER 1

13 Diagram 4 shows two vector, .

x
A(4, 3)

O y

-5 B

Diagram 4

Express :

a) in term of b) in the form

2 Marks

15
14 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

PAPER 2 (2006)

5. Diagram 5 shows a trapezium ABCD.

B C

E
A D

Diagram 5

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