Vectors(^) in
(^) two
(^) dimensions
P1^
8
In Example 8.3 the signs looked after themselves. The component in the i
direction came out positive, that in the j direction negative, as must be the case for
a direction in the fourth quadrant (270° < θ < 360°). This will always be the case
when the conversion is from magnitude−direction form into component form.
The situation is not quite so straightforward when the conversion is carried out
the other way, from component form to magnitude−direction form. In that case,
it is best to draw a diagram and use it to see the approximate size of the angle
required. This is shown in the next example.
ExamPlE 8.4 Write − 5 i + 4 j in magnitude−direction form.
SOlUTION
In this case, the magnitude r = (^5422) +
= 41
= 6.40 (to 2 decimal places).
The direction is given by the angle θ in figure 8.7, but first find the angle α.
tan α = 45 ⇒ α = 38.7° (to nearest 0.1°)
so θ = 180 − α = 141.3°
The vector is (6.40, 141.3°) in magnitude−direction form.
–5i
length 5
4 j
length 4
O
α
i
j
r
θ
Figure 8.7