• # question_answer 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)  $\frac{1}{2},\frac{1}{\sqrt{2}},\frac{1}{2}$

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