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NCERT Class XII Mathematics Chapter - - Solutions

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Question : 2 of 78
Marks: +1, -0
Find the direction cosines of a line which makes equal angles with the coordinate axes.
Solution:  
Let the line makes an angle a with each of the three coordinate axes, then its d.c. are ⟨cos⁡α,cos⁡α,cos⁡α⟩\langle \cos\alpha, \cos\alpha, \cos\alpha \rangle
Also, cos⁡2α+cos⁡2α+cos⁡2α=1\cos^2\alpha + \cos^2\alpha + \cos^2\alpha = 1
⇒3cos⁡2α=1\Rightarrow 3 \cos^2\alpha = 1
⇒cos⁡α=±13\Rightarrow \cos\alpha = \pm \frac{1}{\sqrt{3}}
∴ d.c. of the line are either
⟨13,13,13⟩\langle \frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}} \rangle or ⟨−13,−13,−13⟩\langle -\frac{1}{\sqrt{3}}, -\frac{1}{\sqrt{3}}, -\frac{1}{\sqrt{3}} \rangle
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