A space vector makes the angles \(150^{\circ}\) and \(60^{\circ}\) with the positive direction of \(x\)-and \(y\)-axes. The angle made by the vector with the positive direction \(z\)-axis is

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