Question

Write a unit vector making equal acute angles with the coordinates axes.

Answer 1

Suppose $\overrightarrow{r}$ makes an angle $\alpha$ with each of the axis ${\rm O}{\rm X}$,${\rm O}{\rm Y}$ and ${\rm O}{\rm Z}$.
Then, its direction cosines are $l=\cos \alpha ,m=\cos \alpha ,n=\cos \alpha$.
Now,
$$\begin{gathered} {{\rm I}}^2+m^2+n^2=1 \\ \Rightarrow l^2+l^2+l^2=1\left[\because l=m=n\right] \\ \Rightarrow 3l^2=1 \\ \Rightarrow l^2=\frac{1}{3} \\ \Rightarrow l=\pm \frac{1}{\sqrt{3}} \end{gathered}$$
Since, the angle is acute Hence, we take only positive value
Therefore, unit vector is $\left(\frac{1}{\sqrt{3}}\hat{i}+\frac{1}{\sqrt{3}}\hat{j}+\frac{1}{\sqrt{3}}\hat{k}\right).$