Question

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.

Answer 1

$\cos ^{2} \alpha+\cos ^{2} \beta+\cos ^{2} \gamma=1$
or $\frac{1}{4}+\frac{1}{4}+\cos ^{2} \gamma=1$
or $ \cos ^{2} \gamma=1-\frac{1}{2}=\frac{1}{2}$
$\therefore \cos \gamma=\pm \frac{1}{\sqrt{2}}$
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 $\frac{1}{2}, \frac{1}{2}-\frac{1}{\sqrt{2}}$.