Question

If a line a makes angles $\alpha ,\beta ,\gamma$ with x, y and z axes respectively such that $\alpha +\beta =\frac{\mathrm{\pi }}{2}$, then $\gamma$ = _____________ .

Answer 1

Given; a line makes angles α, β and γ with x, y and z axes respectively.
So, Let us suppose cosα, cosβ and cosγ represent direction cosines

$$\begin{gathered} \mathrm{i}.\mathrm{e}{\cos }^2\alpha +{\cos }^2\beta +{\cos }^2\gamma =1...\left(1\right) \\ \mathrm{Since}\alpha +\beta =\frac{\mathrm{\pi }}{2} \\ \mathrm{i}.\mathrm{e}\beta =\frac{\mathrm{\pi }}{2}-\alpha \\ \mathrm{i}.\mathrm{e}\cos \beta =\cos \left(\frac{\mathrm{\pi }}{2}-\alpha \right) \\ \mathrm{i}.\mathrm{e}\cos \beta =\sin \alpha \\ \mathrm{i}.\mathrm{e}{\cos }^2\beta ={\sin }^2\alpha \\ \therefore \mathrm{equation}\left(1\right)\mathrm{reduces}\mathrm{to}, \\ {\cos }^2\alpha +{\sin }^2\alpha +{\cos }^2\gamma =1 \\ \mathrm{i}.\mathrm{e}1+{\cos }^2\gamma =1 \\ \mathrm{i}.\mathrm{e}{\cos }^2\gamma =0 \\ \mathrm{i}.\mathrm{e}\gamma =\frac{\mathrm{\pi }}{2} \end{gathered}$$