Question

lf a line makes angles $\alpha ,\beta ,\gamma$ with $OX,OY,OZ$ respectively where $O=(0,0,0)$, then $\cos 2\alpha +\cos 2\beta +\cos 2\gamma =$

• A. $1$
• B. $0$
• C. $2$
• D. $-1$

Answer 1

Answer: (D)

$-1$

$(\cos \alpha {)}^2+(\cos \beta {)}^2+(\cos \gamma {)}^2=1$
$2\times \left[\right(\cos \alpha {)}^2+(\cos \beta {)}^2+(\cos \gamma {)}^2]=2$
$2\times \left[\right(\cos \alpha {)}^2+(\cos \beta {)}^2+(\cos \gamma {)}^2]-3=-1$
$2(\cos \alpha {)}^2-1+2(\cos \beta {)}^2-1+2(\cos \gamma {)}^2-1]=-1$
$\left(\cos 2\alpha \right)+\left(\cos 2\beta \right)+\left(\cos 2\gamma \right)=-1$