Cambridge International AS and A Level Mathematics Pure Mathematics 1

Vectors

P1^

8

The magnitude of a is given by the length a in figure 8.4.

a =  4222 + (using Pythagoras’ theorem)

= 4.47   (to    3    significant figures)

The direction is given by the angle θ.

tan.θ= =

2

4 05

θ = 26.6°   (to     3    significant figures)

The vector a is (4.47, 26.6°).

The magnitude of a vector is also called its modulus and denoted by the symbols

| |. In the example a =     4 i +   2 j, the modulus of a, written | a |, is 4.47. Another

convention for writing the magnitude of a vector is to use the same letter, but in

italics and not bold type; thus the magnitude of a may be written a.

ExamPlE 8.2 Write the vector (5, 60°) in component form.

SOlUTION

In the right-angled triangle OPX

OX =    5    cos 60° = 2.5

XP =    5    sin 60° = 4.33

(to     2    decimal places)

O

→

P is

25

433

.

.









or 2.5i + 4.33j.

This technique can be written as a general rule, for all values of θ.

(r, θ) →

r

r

cos

sin

θ

θ







^ =^ (r^ cos^ θ)i^ +^ (r^ sin^ θ)j

ExamPlE 8.3 Write the vector (10, 290°) in component form.

SOlUTION

In this case r =    10   and θ = 290°.

(10, 290°) →

10 290

10 290

342

940

cos

sin

.

–.

°

°









=









to  2    decimal places.

This may also be written 3.42i − 9.40j.

j

i X

P

O

60°

5

Figure 8.5



2ƒ

Figure 8.6