Question

If a line makes angles 90° and 60° respectively with the positive directions of x and y axes, find the angle which it makes with the positive direction of z-axis.

Answer 1

Let the direction cosines of the line be l, m and n.
We know that l² + m² + n² = 1.
Let the line make angle θ with the positive direction of the z-axis.

\[\alpha = 90° \beta = 60°, \gamma = \theta\]

\[\text{ So } , \cos^2 90° + \cos^2 60° + \cos^2 \theta = 1\]

\[ \Rightarrow 0 + \left( \frac{1}{2} \right)^2 + \cos^2 \theta = 1\]

\[ \Rightarrow \cos^2 \theta = 1 - \frac{1}{4}\]

\[ \Rightarrow \cos^2 \theta = \frac{3}{4}\]

\[ \Rightarrow \cos\theta = \pm \frac{\sqrt{3}}{2}\]

\[ \Rightarrow \theta = 30° \text{ or } 150°\]