Question

If $\cos \alpha ,\cos \beta ,\cos \gamma$ are the direction cosines of a vector $\vec{a}$, then $\cos 2\alpha ,\cos 2\beta ,\cos 2\gamma$ is equal to

• A. 2
• B. 3
• C. -1
• D. 0

Answer 1

Answer: (C)

Given, $\cos \alpha ,\cos \beta$ and $\cos \gamma$ are direction cosines of
a vector $a$.
$\therefore {\cos }^2\alpha +{\cos }^2\beta +{\cos }^2\gamma =1$
$\Rightarrow \frac{1+\cos 2\alpha }{2}+\frac{1+\cos 2\beta }{2}+\frac{1+\cos 2y}{2}=1$
$\left[\because 1+\cos 2\theta =2{\cos }^2\theta \right]$
$\Rightarrow \cos 2\alpha +\cos 2\beta +\cos 2\gamma =2-3=-1$