Question

If $\alpha ,\beta ,\gamma$ be the direction angles of a vector and $\cos \alpha =\frac{14}{15},\cos \beta =\frac{1}{3}$ then $\cos \gamma =$

• A. $\pm \frac{2}{15}$
• B. $\frac{1}{5}$
• C. $\pm \frac{1}{15}$
• D. None of these

Answer 1

The correct option is A $\pm \frac{2}{15}$

$\alpha =\frac{\pi }{4}={45}^{\circ },\beta =\frac{\pi }{3}={60}^{\circ }{\cos }^2\alpha +{\cos }^2\beta +{\cos }^2\gamma =1{\cos }^2{45}^{\circ }+{\cos }^2{60}^{\circ }+{\cos }^2\gamma =1.5+.25+{\cos }^2\gamma =1{\cos }^2\gamma =1-.75=.25\cos \gamma =\pm .5\Rightarrow \gamma ={\cos }^{-1}\left(.5\right)=\frac{\pi }{3}\Rightarrow \gamma ={\cos }^{-1}(-.5)=\frac{2\pi }{3}$

So options $A$ and $B$ are correct.