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 $-\frac{1}{2},\frac{1}{\sqrt{2}},\frac{1}{2},$