Question

A vector makes an angle of $\frac{\mathrm{\pi }}{4}$ with each of x-axis and y-axis. Find the angle made by it with the z-axis.

Answer 1

Let the vector $\overrightarrow{OP}$ makes an angle $\alpha =45°\mathrm{and}\beta =45°$ with $OX,OY$ respectively. Suppose $\overrightarrow{OP}$ is inclined at angle $\gamma$ to $OZ$.
Let $l,m,n$ be the direction cosines of $\overrightarrow{OP}$. Then,
$l=\cos \frac{\mathrm{\pi }}{4}=\frac{1}{\sqrt{2}}$ , $m=\cos \frac{\mathrm{\pi }}{4}=\frac{1}{\sqrt{2}}$ and $n=\cos \gamma$
Now, we have,

$$\begin{gathered} l^2+m^2+n^2=1 \\ \Rightarrow \frac{1}{2}+\frac{1}{2}+n^2=1 \\ \Rightarrow n^2=0 \\ \Rightarrow n=0 \\ \Rightarrow \cos \gamma =\cos \frac{\mathrm{\pi }}{2} \\ \Rightarrow \gamma =\frac{\mathrm{\pi }}{2} \end{gathered}$$
Hence, the angle made by it with the $z-$axis is $\frac{\mathrm{\pi }}{2}$.