Question

A space vector makes the angles $150^{\circ }$ and $60^{\circ }$ with the positive direction of $X-$ and $Y-$axis. The angle made by the vector with the positive direction of $Z-$axis is .............

• A. $180^{\circ }$
• B. $120^{\circ }$
• C. $90^{\circ }$
• D. $60^{\circ }$

Answer 1

Answer: (C)

We know that, the condition when a space vector makes the angles $\alpha ,\beta$ and $\gamma$ with the positive direction of $x,y$ and $z$ -axes respectively is
${\cos }^2\alpha +{\cos }^2\beta +{\cos }^2\gamma =1$...(i)
Given that, $\alpha =150^{\circ },\beta =60^{\circ },\gamma =?$
From Eq (i), ${\cos }^{2}150^{\circ }+{\cos }^{2}60^{\circ }+{\cos }^2\gamma =1$
$\left({\sin }^{2}60^{\circ }+{\cos }^{2}60^{\circ }\right)+{\cos }^2\gamma =1$
$1+{\cos }^2\gamma =1$
$\Rightarrow {\cos }^2\gamma =0$
$\Rightarrow \cos \gamma =0=\cos 90^{\circ }$
$\Rightarrow \gamma =90^{\circ }$