Question

A vector makes an angle of \[\frac{\pi}{4}\] 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 \[\alpha = 45^{\circ} \text{ and }\beta = 45^{\circ}\] 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,
\[l = \cos \frac{\pi}{4} = \frac{1}{\sqrt{2}}\]
\[m = \cos \frac{\pi}{4} = \frac{1}{\sqrt{2}}\]
\[n = \cos \gamma\]
Now, we have,
\[l^2 + m^2 + n^2 = 1\]
\[ \Rightarrow \frac{1}{2} + \frac{1}{2} + n^2 = 1\]
\[ \Rightarrow n^2 = 0\]
\[ \Rightarrow n = 0\]
\[ \Rightarrow \cos \gamma = \cos \frac{\pi}{2}\]
\[ \Rightarrow \gamma = \frac{\pi}{2}\]
Hence, the angle made by it with the \[z -\] axis is \[\frac{\pi}{2}\]