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प्रश्न
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.
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उत्तर
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.
\[\alpha = 90° \beta = 60°, \gamma = \theta\]
\[\text{ So } , \cos^2 90° + \cos^2 60° + \cos^2 \theta = 1\]
\[ \Rightarrow 0 + \left( \frac{1}{2} \right)^2 + \cos^2 \theta = 1\]
\[ \Rightarrow \cos^2 \theta = 1 - \frac{1}{4}\]
\[ \Rightarrow \cos^2 \theta = \frac{3}{4}\]
\[ \Rightarrow \cos\theta = \pm \frac{\sqrt{3}}{2}\]
\[ \Rightarrow \theta = 30° \text{ or } 150°\]
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