Question

If a line makes an angle with $\frac{\mathrm{\pi }}{4}$ each of y and z-axis, then the angle which it makes with x-axis, is _____________.

Answer 1

Since a line makes an angle $\frac{\mathrm{\pi }}{4}$ with y and z-axis
Then l = cos$\alpha$, m = cos$\frac{\mathrm{\pi }}{4}$, n = cos$\frac{\mathrm{\pi }}{4}$
Since l² + m² + n² = 1
$$\begin{gathered} \mathrm{i}.\mathrm{e}{\cos }^2\alpha +{\left(\frac{1}{\sqrt{2}}\right)}^2+{\left(\frac{1}{\sqrt{2}}\right)}^2=1 \\ \mathrm{i}.\mathrm{e}{\cos }^2\alpha +\frac{1}{2}+\frac{1}{2}=1 \\ \mathrm{i}.\mathrm{e}{\cos }^2\alpha =0 \\ \mathrm{i}.\mathrm{e}\cos \alpha =0 \\ \mathrm{i}.\mathrm{e}\alpha =\frac{\mathrm{\pi }}{2} \end{gathered}$$
∴ The angle it makes with x-axis is $\frac{\mathrm{\pi }}{2}$.