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 \,2 y}{2}=1 $
${\left[\because 1+\cos \,2 \theta=2 \cos ^{2} \theta\right]} $
$\Rightarrow \cos \,2 \alpha+\cos \,2 \beta+\cos \,2 \gamma=2-3=-1$