Answer
$v=7[\cos 64.6^{\circ} i+\cos 149.0^{\circ} j+\cos 106.6^{\circ}k ]$
Work Step by Step
The magnitude of any vector (let us say $v$) can be determined using the formula
$||v||=\sqrt{p^2+q^2+r^2} $
Let $\alpha, \beta, \gamma $ be the three direction angles with the following formulas:
$\alpha =\cos^{-1} \dfrac{p}{||v||}$ and $\beta =\cos^{-1} \dfrac{q}{||v||}$ and $\gamma =\cos^{-1} \dfrac{r}{||v||}$
In our case:
$||v||=\sqrt {(3)^2+(-6)^2+(-2)^2}=\sqrt {49}=7$ and $p=3; q=-6 ; r=-2$
Therefore,
$\alpha =\cos^{-1} \dfrac{3}{7} \approx 64.6^{\circ} $ and $\beta =\cos^{-1} \dfrac{-6}{7} \approx 149.0^{\circ} $ and $\gamma =\cos^{-1} \dfrac{2}{7}\approx 106.6^{\circ}$
Thus, we get
$v=||v|| [\cos \alpha i+ \cos \beta j+ \cos \gamma k ] =7[\cos 64.6^{\circ} i+\cos 149.0^{\circ} j+\cos 106.6^{\circ}k ]$
