Question

If a vector $\overline{x}$ makes angles of measures $\frac{2\pi }{3}$ and $\frac{\pi }{3}$ with Y - axis and Z - axis respectively then find the measure of the angle made by $\overline{x}$ with X - axis.

Answer 1

If angle with $X,Y$ and $Z-axis$ be $\alpha ,\beta$ and $\gamma$

Then,

${\cos }^2\alpha +{\cos }^2\beta +{\cos }^2\gamma =1$

$⟹{\cos }^2\alpha +{\cos }^2\frac{2\pi }{3}+{\cos }^2\frac{\pi }{3}=1$

$⟹{\cos }^2\alpha +\frac{1}{4}+\frac{1}{4}=1$

$⟹\cos \alpha =\pm \frac{1}{\sqrt{2}}$

Hence, $\alpha =\frac{\pi }{4}$ or $\alpha =\frac{3\pi }{4}$