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

Answer: 2

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