Question

Given vector u equals 30 (cos 60 degrees, sin 60) what are the magnitude and direction of −4u?

Answer 1

ANSWER:

Magnitude is -120 and the direction is 60°

STEP-BY-STEP EXPLANATION:

The unit vector is:

$u=30\langle\cos60\degree,\sin60\degree\rangle$

Let's call the new vector v, which would be:

v = -4u

The magnitude is calculated as follows:

$\begin{gathered} ||u||=30\cdot\sqrt{\left(cos\:60\right)^2+\left(sin\:60\right)^2}=30 \\ ||v||=-4\cdot||u||=-4\cdot30=-120 \\ ||v||=-120 \end{gathered}$

For the direction:

$\begin{gathered} \tan\theta=\frac{-120\cdot\sin60}{-120\cdot\cos60}=\frac{\frac{\sqrt{3}}{2}}{\frac{1}{2}} \\ \tan\theta=\sqrt{3} \\ \theta=\tan^{-1}(\sqrt{3}) \\ \theta=60\degree \end{gathered}$

Which means that the magnitude is -120 and the direction is 60°