1. What is a Vector? AB and CD have the same distance
However, they have different directions B
From A to B, AB
From C to D, CD
A Vector is a quantity that has both size and direction. A C D
Therefore AB and CD are different vectors despite having the same size

2. Vector Notation PQ = ( ) GH = ( ) RW = ( ) FK = ( ) 2 4 4 2 3 -2 -4 -2

3. ( ) ( ) ( ) ( ) Addition of Vectors RT = ( ) TS = ( ) RT + TS = RS5 S -2
TS = ( ) -1 6
You can add these column vectors to find RS by adding the corresponding points R T
The diagram shows
RT + TS = RS
( )
( )
( )
( ) -1 4 5 + = = -2 6 4

4. ( ) + ( ) = ( ) ( ) + ( ) = ( ) ( ) + ( ) = ( ) AB + BC = AC 1 2 3 S A
( ) + ( ) = ( ) 1 2 3 S A -1 -3 2 C R
RS ST = RT T
( ) + ( ) = ( ) 3 1 4 B 1 -2 -1 Z X
XY YZ = XZ
( ) + ( ) = ( ) -1 -5 -4 1 -3 4 Y

5. B
can also be written as a small letter
The vector AB a A
If vectors are the same size and have the same direction then they can be labelled with the same letter a a
If there are two vectors of the same size but one is going in the opposite direction you can indicate this by putting a negative sign in front of the letter a -a

6. Subtraction of Vectors
( ) 3
p = -4 -4
( )
So we get p + -q or p - q
q = -5
p - q
To find p – q subtract the corresponding points 3 -4
( )
( )
p - q = - -4 -5 q -q p
( )
3 - -4 =
( ) 7 = 1

7. Multiplying a Vector by a Scalar
The diagram shows that the vector has been multiplied by a scalar 2 to get a 2a 3
a = ( ) 3 a 6
2a = ( ) = ( )
2 x 3 2a
- ½ a
2 x 3 6 a
The vector has been multiplied by a scalar
- ½
This means that - ½ a is half as long as a and in the opposite direction.
- ½ a = ( ) = ( )
- ½ x 3
-1 ½
So we get …..
- ½ x 3
-1 ½

8. 2s - t -3t - ½ s s – t 3t – s -4(2t – 3s) -3 6 t = ( ) s = ( ) 4 -2
Write the following as column Vectors: 2s
- t
-3t
- ½ s
s – t
3t – s
-4(2t – 3s)

9. Find the Vector in terms of p and q

10. The vector PQ =
q – p Q
Finding the midpoint of PQ …… M q
OM = p + ½ PQ P
= p + ½ (q – p) p
= p + ½ q – ½ p O
= ½ p + ½ q
= ½ (p + q)

11. B M
OA = 4p + 3q and OB = 3p -2q A
Find AB in terms of p and q
AB = - (4p + 3q) + 3p – 2q O
AB = -4p – 3q + 3p – 2q
AB = -p - 5q
Find the position vector of the midpoint of AB
Position vector of midpoint of AB is ½ (a + b)
= ½ [(4p + 3q) + (3p – 2q)]
= ½ [ 7p + q ]

12. The diagram P divides the line ST in the ratio 3 : 1
Find SP in terms of ST
SP = ¾ ST
Find SP in terms of s and t
SP = ¾ (t – s) O S
Show that SP = ¼ (s + 3t)
SP = s + ¾ (t – s)
SP = ¼ s + ¾ t
SP = ¼ (s + 3t)

13. C F G B A M K E H N J L