Cambridge International AS and A Level Mathematics Pure Mathematics 1

Vectors

P1^

8

The next example shows you how to use it to find the angle between two vectors

given numerically.

ExamPlE 8.11 Find the angle between the vectors ^34 

and (^) – 125 .
SOlUTION
Let a=^34  ⇒ | a | = 3422 + = 5
and b=– 125  ⇒ | b | = 5122 +(– 2 ) = 13.
The scalar product
3
4

5

12















. – 

= 3 × 5 + 4 × (−12)

= 15 − 48

= −33.

Substituting in a. b = | a | | b | cos θ gives

− 33     =  5    ×  13   × cos θ

cosθ=–^33

65

⇒ θ = 120.5°.

Perpendicular vectors

Since cos 90° = 0, it follows that if vectors a and b are perpendicular then

a. b = 0.

Conversely, if the scalar product of two non-zero vectors is zero, they are

perpendicular.

ExamPlE 8.12 Show that the vectors aa=







2

4

and bb=







6

– 3

are perpendicular.

SOlUTION

The scalar product of the vectors is

aa..bb=















2

4

6

3

.

–

= 2 × 6 + 4 × (−3)

= 12 − 12 = 0.

Therefore the vectors are perpendicular.