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

Answer 1

The correct option is B $\frac{1}{\sqrt{2}},\frac{1}{2},\frac{1}{2}$

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 $\beta =\frac{\pi }{3}$
We know that, ${\cos }^2\alpha +{\cos }^2\beta +{\cos }^2\gamma =1$
$\therefore {\cos }^2\frac{\pi }{4}+{\cos }^2\frac{\pi }{3}+{\cos }^2\gamma =1$
$\Rightarrow {\left(\frac{1}{\sqrt{2}}\right)}^2+{\left(\frac{1}{2}\right)}^2+{\cos }^2\gamma =1$
$\Rightarrow \frac{1}{2}+\frac{1}{4}+{\cos }^2\gamma =1$
$\Rightarrow {\cos }^2\gamma =\frac{1}{4}$
$\Rightarrow \cos \gamma =\frac{1}{2}$
$\therefore \gamma =\frac{\pi }{3}$
Hence, direction cosines are $\cos \alpha ,\cos \beta ,\cos \gamma$
i.e. $\frac{1}{\sqrt{2}},\frac{1}{2},\frac{1}{2}$.