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# Solution for A Line Makes an Angle of 60° with Each of X-axis and Y-axis. Find the Acute Angle Made by the Line with Z-axis. - CBSE (Science) Class 12 - Mathematics

ConceptDirection Cosines and Direction Ratios of a Line

#### Question

A line makes an angle of 60° with each of X-axis and Y-axis. Find the acute angle made by the line with Z-axis.

#### Solution

$\text { It is given that a line makes an angle of 60° with both x - axis and y - axis } .$

$\text{ Suppose the line makes an angle of } \alpha \text{ with the z - axis }.$

$\Rightarrow l = cos\ 60° = \frac{1}{2}$

$m = \cos 60° = \frac{1}{2}$

$n = \cos \alpha$

$\text{ We know }$

$l^2 + m^2 + n^2 = 1$

$\Rightarrow \left( \frac{1}{2} \right)^2 + \left( \frac{1}{2} \right)^2 + \left( \cos \alpha \right)^2 = 1$

$\Rightarrow \frac{1}{4} + \frac{1}{4} + \cos {}^2 \alpha = 1$

$\Rightarrow \cos \alpha = \frac{1}{\sqrt{2}}$

$\Rightarrow \alpha = 45°$

$\text{ Thus, the line makes an angle of } 45° \text{ with the z - axis }.$

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#### Video TutorialsVIEW ALL [3]

Solution A Line Makes an Angle of 60° with Each of X-axis and Y-axis. Find the Acute Angle Made by the Line with Z-axis. Concept: Direction Cosines and Direction Ratios of a Line.
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