Question

If a vector $\overrightarrow{x}$ makes angles with measure $\frac{\pi }{4}$ and $\frac{5\pi }{4}$ with positive directions of X-axis and Y-axis respectively, then $\overrightarrow{x}$ made angle of measure ________ with positive direction of Z-axis.

• A. $\frac{5\pi }{4}$
• B. $\frac{\pi }{3}$
• C. $\frac{\pi }{2}$
• D. $\frac{\pi }{4}$

Answer 1

The correct option is C $\frac{\pi }{2}$

If $\overrightarrow{x}$ makes angles with measure $\frac{\pi }{4}$ and $\frac{5\pi }{4}$ with positive directions of X-axis and Y-axis respectively,
$\alpha =\frac{\pi }{4},\beta =\frac{5\pi }{4}$
Let angle with positive direction of Z-axis is $\gamma$
Then ${\cos }^2\alpha +{\cos }^2\beta +{\cos }^2\gamma =1$
$\Rightarrow \frac{1}{2}+\frac{1}{2}+{\cos }^2\gamma =1\Rightarrow \cos \gamma =0\Rightarrow \gamma =\frac{\pi }{2}$