Question

Show that the vector $\hat{i}+\hat{j}+\hat{k}$ is equally inclined to the axes OX,OY,and OZ.

Answer 1

Let $\vec{a}=\hat{i}+\hat{j}+\hat{k}$
Then,
$|\vec{a}|=\sqrt{1^2+1^2+1^2}=\sqrt{3}$
Therefore,the direction cosines of $\vec{a}$ are$(\frac{1}{\sqrt{3}},\frac{1}{\sqrt{3}},\frac{1}{\sqrt{3}}).$
Now,let $α,β$,and $γ$ be the angles formed by $\vec{a}$ with the positive directions $x,y$,and $z$ axes.
Then,we have $cosα=\frac{1}{\sqrt{3}},cosβ=\frac{1}{\sqrt{3}},cosγ=\frac{1}{\sqrt{3}}.$
Hence,the given vector is equally inclined to axes OX,OY,and OZ.