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
2
Figure 8.6