Question

A line makes an angle of 60° with each of X-axis and Y-axis. Find the acute angle made by the line with Z-axis.

Answer 1

$$\begin{gathered} \mathrm{It}\mathrm{is}\mathrm{given}\mathrm{that}\mathrm{a}\mathrm{line}\mathrm{makes}\mathrm{an}\mathrm{angle}\mathrm{of}60°\mathrm{with}\mathrm{both}x-\mathrm{a}\mathrm{x}\mathrm{i}\mathrm{s}\mathrm{and}y-\mathrm{a}\mathrm{x}\mathrm{i}\mathrm{s}. \\ \mathrm{Suppose}\mathrm{the}\mathrm{line}\mathrm{makes}\mathrm{an}\mathrm{angle}\mathrm{of}\alpha \mathrm{with}\mathrm{the}z-\mathrm{axis}. \\ \Rightarrow l=\cos 60°=\frac{1}{2} \\ m=\cos 60°=\frac{1}{2} \\ n=\cos \alpha \\ \mathrm{We}\mathrm{know} \\ {l}^{\mathit{2}}+\mathit{}{m}^{\mathit{2}}+n^{\mathit{2}}=1 \\ \Rightarrow {\left(\frac{1}{2}\right)}^2+{\left(\frac{1}{2}\right)}^2+{\left(\cos \alpha \right)}^2=1 \\ \Rightarrow \frac{1}{4}+\frac{1}{4}+\cos {}^2\alpha =1 \\ \Rightarrow \cos \alpha =\frac{1}{\sqrt{2}} \\ \Rightarrow \alpha =45° \\ \mathrm{Thus},\mathrm{the}\mathrm{line}\mathrm{makes}\mathrm{an}\mathrm{angle}\mathrm{of}45°\mathrm{with}\mathrm{the}z-\mathrm{axis}. \end{gathered}$$