Question

If a vector $\overrightarrow{A}$ makes angles $\alpha ,\beta$ and $\gamma$ with X, Y and Z axes respectively then ${\sin }^2\alpha +{\sin }^2\beta +{\sin }^2\gamma =$

• A. $0$
• B. $1$
• C. $2$
• D. $3$

Answer 1

The correct option is C $2$

$\text{Step 1: Use direction cosines equation}$

If $\cos \alpha ,\cos \beta$ and $\cos \gamma$ be the direction cosines of a vector, then

${\cos }^2\alpha +{\cos }^2\beta +{\cos }^2\gamma =1$ $....\left(1\right)$

$\text{Step 2: Solving equation}$

We know,

${\sin }^2\theta +{\cos }^2\theta =1$

$\therefore {\cos }^2\alpha +{\sin }^2\alpha +{\cos }^2\beta +{\sin }^2\beta +{\sin }^2\gamma +{\cos }^2\gamma =1+1+1$

From eqn $\left(1\right)$

${\sin }^2\alpha +{\sin }^2\beta +{\sin }^2\gamma +1=3$

$\Rightarrow$ ${\sin }^2\alpha +{\sin }^2\beta +{\sin }^2\gamma =2$

Hence, option $C$ is correct.