Question

A unit vector \[\overrightarrow{r}\] makes angles \[\frac{\pi}{3}\] and \[\frac{\pi}{2}\] with \[\hat{j}\text{ and }\hat{k}\]  respectively and an acute angle θ with \[\hat{i}\]. Find θ.

Answer 1

A unit vector makes an angle \[\frac{\pi}{3}\] and \[\frac{\pi}{2}\] with \[\hat{j}\text{ and }\hat{k}\]
Let \[I, m, n\] be its direction cosines
\[l = \cos\theta , m = \cos\left( \frac{\pi}{3} \right) = \frac{1}{2} , n = \cos\left( \frac{\pi}{2} \right) = 0\]
Now
∴ \[l^2 + m^2 + n^2 = 1\]
\[\Rightarrow\] \[l^2 + \frac{1}{4} + 0 = 1\]
\[\Rightarrow\] \[l^2 = 1 - \frac{1}{4} = \frac{3}{4}\]
\[\Rightarrow\] \[l = \pm \frac{\sqrt{3}}{2}\]
∴ \[\vec{r}\] makes an angle \[30^{\circ} , 150^{\circ}\] with \[\stackrel\frown{i}\] 
Since, angle θ is acute.
∴ \[\theta = 30^{\circ}\]