Discussion Forum

If a line segment  OP makes angles of   $\frac{\pi}{4}$ and  $\frac{\pi}{3}$  with X-axis and Y-axis,  respectively . Then, the direction cosines are 

Answer:

Explanation:

Let  $\alpha,\beta$ and $\gamma$ be the angles made by the line segment OP with X-axis, Y-axis and Z-axis , respectively

 Given:  $\alpha=\frac{\pi}{4}$ and $\beta=\frac{\pi}{3} $

We know that  $\cos^{2}\alpha+\cos^{2}\beta+\cos^{2}\gamma=1 $

 $\therefore$    $\cos^{2}\frac{\pi}{4}+\cos^{2}\frac{\pi}{3}+\cos^{2}\gamma=1 $

$\Rightarrow \left(\frac{1}{\sqrt{2}}\right)^{2}+\left(\frac{1}{2}\right)^{2}+\cos^{2}\gamma=1$

$\Rightarrow \frac{1}{2}+\frac{1}{4}+\cos^{2}\gamma=1$

$\Rightarrow \cos^{2}\gamma=\frac{1}{4}$

$\Rightarrow \cos^{}\gamma=\frac{1}{\sqrt{2}}$

$\therefore$      $\gamma=\frac{\pi}{4}$

Hence , direction  cosines are  $\cos\alpha,\cos\beta,\cos\gamma$   

 i.e,  $\frac{1}{\sqrt{2}},\frac{1}{2},\frac{1}{\sqrt{2}}$