Question

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

• A. $2$
• B. $3$
• C. $-1$
• D. $0$

Answer 1

The correct option is C $-1$

Since $\cos \alpha ,\cos \beta ,\cos \gamma$ are direction cosines of vector $\overrightarrow{a},$
${\cos }^2\alpha +{\cos }^2\beta +{\cos }^2\gamma =1$
$\text{Now,}\cos 2\alpha +\cos 2\beta +\cos 2\gamma$
$=(2{\cos }^2\alpha -1)+(2{\cos }^2\beta -1)+(2{\cos }^2\gamma -1)$
$=2({\cos }^2\alpha +{\cos }^2\beta +{\cos }^2\gamma )-3$
$=2\times 1-3=-1$