Question

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

• A. $\frac{1}{\sqrt{2}}, \frac{\sqrt{3}}{2}, \frac{1}{\sqrt{2}}$
• B. $\frac{1}{\sqrt{2}}, \frac{1}{2}, \frac{1}{\sqrt{2}}$
• C. $1, \sqrt{3}, 1$
• D. $1, \frac{1}{\sqrt{3}}, 1$

Answer 1

Answer: (B)

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

Answer 2

Let $\alpha,\,\beta$ and $\gamma$ be the angles made by the line segment OP with X-axis, Y-axis and Z-axis, respectively.
Given : $\alpha = \frac{\pi}{4} and \beta = \frac{\pi }{3}$
$\because cos^{2} \alpha + cos^{2} \beta+ cos^{2} \gamma = 1$
$\therefore cos^{2} \frac{\pi }{4} + cos^{2} \frac{\pi }{3} +cos^{2} \gamma = 1$
$\Rightarrow \left(\frac{1}{\sqrt{2}}\right)^{2} +\left(\frac{1}{2}\right)^{2} +cos^{2} \gamma = 1$
$\Rightarrow \frac{1}{2}+\frac{1}{4} +cos^{2} \gamma = 1$
$\Rightarrow cos^{2} \gamma = \frac{1}{4}$
$\Rightarrow cos^{2} \gamma = \frac{1}{\sqrt{2}}$
$\therefore \gamma = \frac{\pi }{4}$
Hence, direction cosines are $cos \,\alpha,\, cos \,\beta, \,cos \,\gamma$
$i.e \,\frac{1}{\sqrt{2}}, \frac{1}{2}, \frac{1}{\sqrt{2}}.$