Question

A line makes an obtuse angle with the positive x-axis and angles $ \frac{\pi }{4} $ and $ \frac{\pi }{3} $ with the positive $y$ and $z$ axes respectively. Its direction cosine are

• A. $ \frac{1}{\sqrt{2}},\frac{1}{\sqrt{2}},\frac{1}{2} $
• B. $ \frac{1}{\sqrt{2}},-\frac{1}{2},\frac{1}{2} $
• C. $ -\frac{1}{2},\frac{1}{\sqrt{2}},\frac{1}{2} $
• D. $ \frac{1}{2},\frac{1}{\sqrt{2}},\frac{1}{2} $

Answer 1

Answer: (C)

Let $ m=\cos \frac{\pi }{4}=\frac{1}{\sqrt{2}} $
and $ n=\cos \frac{\pi }{3}=\frac{1}{2} $
$ \because $ $ {{l}^{2}}+{{m}^{2}}+{{n}^{2}}=1 $
$ \Rightarrow $ $ l=\sqrt{1-({{m}^{2}}+{{n}^{2}})}=\sqrt{1-\left( \frac{1}{2}+\frac{1}{4} \right)} $
$ =\sqrt{1-\frac{3}{4}} $
$ \Rightarrow $ $ l=\pm \frac{1}{2} $
Since, line makes an obtuse angle, so we angle
$ l=-\frac{1}{2} $
$ \therefore $ Direction cosines are $ -\frac{1}{2},\frac{1}{\sqrt{2}},\frac{1}{2}, $