Question

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

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

Answer 1

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

Let $\alpha ,\beta$ and $\gamma$ be the direction ratios of the vector.

and let $a,b$ and $c$ be the direction angles of the vector.

Then,

Then,

$\alpha =cos\left(a\right)=cos\frac{\pi }{4}=\frac{1}{\sqrt{2}}$

$\beta =cos\left(b\right)=cos\frac{5\pi }{4}=-\frac{1}{\sqrt{2}}$

$\gamma =cos\left(c\right)$

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

$⟹\frac{1}{2}+\frac{1}{2}+{\gamma }^2=1$

$⟹\gamma =0$

$⟹cos\left(c\right)=0$

$⟹c=\frac{\pi }{2}$ or $c=\frac{3\pi }{2}$

Hence, the answer is option (A).