Question

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

Answer 1

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

Then,

${\displaystyle \left|\overrightarrow{a}\right|=\sqrt{1^2+1^2+1^2}=\sqrt{3}}$

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

Now, let α, β, and λ be the angles formed by ${\displaystyle \overrightarrow{a}}$ with the positive directions of the x, y, and z axes.

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

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