Question

If a line makes angles $\frac{\mathrm{\pi }}{2},\frac{3\mathrm{\pi }}{4}\mathrm{and}\frac{\mathrm{\pi }}{4}$ with x, y, z axes respectively, then its direction cosines are _____________.

Answer 1

Since direction cosines are given by
l = cosα, m = cosβ and n = cosγ
where α is angle with x-axis, β is angle with y-axis and γ is angle with z-axis
Given, a line makes angles $\frac{\mathrm{\pi }}{2},\frac{3\mathrm{\pi }}{4}\mathrm{and}\frac{\mathrm{\pi }}{4}$ with x, y and z axes respectively.
$\begin{array}{rcl}\therefore l& =& \cos \frac{\mathrm{\pi }}{2}=0\\ m& =& \cos \frac{3\mathrm{\pi }}{4}=-\frac{1}{\sqrt{2}}\left[\because \cos \frac{3\mathrm{\pi }}{4}=\cos \left(\mathrm{\pi }-\frac{\mathrm{\pi }}{4}\right)\right]\\ n& =& \cos \frac{\mathrm{\pi }}{4}=\frac{1}{\sqrt{2}}\end{array}$