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)

$90^\circ$

Answer 2

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), $\cos ^{2} 150^{\circ}+\cos ^{2} 60^{\circ}+\cos ^{2} \gamma=1$
$\left(\sin ^{2} 60^{\circ}+\cos ^{2} 60^{\circ}\right)+\cos ^{2} \gamma=1$
$1+\cos ^{2} \gamma=1$
$\Rightarrow \cos ^{2} \gamma=0$
$\Rightarrow \cos \gamma=0=\cos 90^{\circ}$
$\Rightarrow \gamma=90^{\circ}$