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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

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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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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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