#### 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.

#### Solution

Let the direction cosines of the line be *l*,* m* and *n*.

We know that *l*^{2} + *m*^{2} + *n*^{2} = 1.

Let the line make angle *θ* with the positive direction of the *z*-axis.

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

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

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

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

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

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

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

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Solution 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. Concept: Direction Cosines and Direction Ratios of a Line.