Question

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

Answer 1

Let $\overrightarrow{a}=\hat{i}+\hat{j}+\hat{k}$

Then, $|\overrightarrow{a}|=\sqrt{1^2+1^2+1^2}=\sqrt{3}$

Therefore, the direction cosines of $\overrightarrow{a}$ are $(\frac{1}{\sqrt{3}},\frac{1}{\sqrt{3}},\frac{1}{\sqrt{3}})$.

Now, let $\alpha ,\beta$, and $\gamma$ be the angles formed by $\overrightarrow{a}$ with the positive direction directions of $x,y$, and $z$ axes.

Then, we have $\cos \alpha =\frac{1}{\sqrt{3}},\cos \beta =\frac{1}{\sqrt{3}},\cos \gamma =\frac{1}{\sqrt{3}}$.

Hence, the given vector is equally inclined to axes $OX,OY$, and $OZ$.