How many unit vectors are there for which $\cos \alpha =\frac{1}{2}$ and $\cos \beta =\frac{1}{2},$, where $\alpha$ and $\beta$ are angles made with X-axis and Y-axis, respectively.

${\cos }^2\alpha +{\cos }^2\beta +{\cos }^2\gamma =1$

$\begin{array}{l}\text{ or }\frac{1}{4}+\frac{1}{4}+{\cos }^2\gamma =1\\ \text{ or }{\cos }^2\gamma =1-\frac{1}{2}=\frac{1}{2}\\ \therefore \cos \gamma =\pm \frac{1}{\sqrt{2}}\end{array}$

Two values of cos $\gamma$ are possible. Hence, two vectors are possible, one corresponding to direction cosines $\frac{1}{2},\frac{1}{2},\frac{1}{\sqrt{2}}$ and other corresponding to direction cosines $\frac{1}{2},\frac{1}{2}-\frac{1}{\sqrt{2}}$