Question

Show that the direction cosines of a vector equally inclined to the axes OX, OY and OZ are $\frac{1}{\sqrt{3}},\frac{1}{\sqrt{3}},\frac{1}{\sqrt{3}}.$

Answer 1

Suppose the vector makes equal angle $\alpha$ with the coordinate axis.
Then, its direction cosines are $l=\cos \alpha ,m=\cos \alpha ,n=\cos \alpha$. Therefore, $l=m=n$.
$$\begin{gathered} l^2+m^2+n^2=1 \\ \Rightarrow l^2+l^2+l^2=1 \\ \Rightarrow 3l^2=1 \\ \Rightarrow l^2=\frac{1}{3} \\ \Rightarrow l=\frac{1}{\sqrt{3}} \end{gathered}$$
Hence, direction cosines are $\frac{1}{\sqrt{3}},\frac{1}{\sqrt{3}},\frac{1}{\sqrt{3}}$.