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Solution for 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. - CBSE (Science) Class 12 - Mathematics

ConceptDirection Cosines and Direction Ratios of a Line

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 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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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.
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