Cambridge International AS and A Level Mathematics Pure Mathematics 1

The angle between two vectors

The angle between two vectors

271

P1^

8

The angle between two vectors

●^  As^ you^ work^ through^ the^ proof^ in^ this^ section,^ make^ a^ list^ of^ all^ the^ results^ that^

you are assuming.

To find the angle θ between the

two vectors

O

→

A = a = a 1 i + a 2 j

and

O

→

B = b = b 1 i + b 2 j

start by applying the cosine rule to

triangle OAB in figure 8.21.

cosθ=OA2O+OAOB–× BAB

22 2

In this, OA, OB and AB are the lengths of the vectors O

→

A, O

→

B and A

→

B, and so

OA = | a | = aa 12 + 22                  and OB = | b | = bb 12 + 22.

The vector A

→

B = b − a    = (b 1 i + b 2 j) − (a 1 i + a 2 j)

= (b 1   − a 1 )i + (b 2     − a 2 )j

and so its length is given by

AB = | b − a | =     (–ba 11 )(^2 +ba 22 –)^2.

Substituting for OA, OB and AB in the cosine rule gives

cos

() ()–[(– )( –)]

θ=

aa++bb+ ba + ba

a

2

1

2

2

2

1

2

21 1

2

22

2

2 2

11

2

2

2

1

2

+×ab +b 2

=

aa 12 ++ 22 bb 12 +^22 (ba^2112 ba 1 +^21 ++ba^2222 ba 2 22 )

2

–– –

aaaabb

This simplifies to

cosθ=

22 +

2

ab 11 ab 22

aabb^

=

ab 11 +ab 22

aabb

The expression on the top line, a 1 b 1  + a 2 b 2 , is called the scalar product (or dot

product) of the vectors a and b and is written a. b. Thus

cos.θ= aa..bb

aabb

This result is usually written in the form

a. b = | a | | b | cos θ.

y

(^2) x
θ
a b
b 1  b 2
a 1  a 2
$
%
Figure 8.21