Question

The direction cosines of a line equally inclined to three mutually perpendicular lines having direction cosines as $l_{1}, m_{1}, n_{1} ; l_{2}, m_{2}, n_{2}$ and $l_{3}, m_{3}, n_{3}$ are

• A. $l_{1}+l_{2}+l_{3}, m_{1}+m_{2}+m_{3}, n_{1}+n_{2}+n_{3}$
• B. $\frac{l_{1}+l_{2}+l_{3}}{\sqrt{3}}, \frac{m_{1}+m_{2}+m_{3}}{\sqrt{3}}, \frac{n_{1}+n_{2}+n_{3}}{\sqrt{3}}$
• C. $\frac{l_{1}+l_{2}+l_{3}}{3}, \frac{m_{1}+m_{2}+m_{3}}{3}, \frac{n_{1}+n_{2}+n_{3}}{3}$
• D. none of these

Answer 1

Answer: (B)

Since the three lines are mutually perpendicular,
$\therefore l_{1} l_{2}+m_{1} m_{2}+n_{1} n_{2}=0$
$ l_{2} l_{3}+m_{2} m_{3}+n_{2} n_{3}=0 $
$l_{3} l_{1}+m_{3} m_{1}+n_{3} n_{1}=0$
Also, $ l_{1}^{2}+m_{1}^{2}+n_{1}^{2} =1, l_{2}^{2}+m_{2}^{2}+n_{2}^{2}=1$, $l_{3}^{2}+m_{3}^{2}+n_{3}^{2} =1$
Now, $\left(l_{1}+l_{2}+l_{3}\right)^{2}+\left(m_{1}+m_{2}+m_{3}\right)^{2}+\left(n_{1}+n_{2}+n_{3}\right)^{2} $
$=\left(l_{1}^{2}+m_{1}^{2}+n_{1}^{2}\right)+\left(l_{2}^{2}+m_{2}^{2}+n_{2}^{2}\right) $
$+\left(l_{3}^{2}+m_{3}^{2}+n_{3}^{2}\right)+2\left(l_{1} l_{2}+m_{1} m_{2}+n_{1} n_{2}\right) $
$+2\left(l_{2} l_{3}+m_{2} m_{3}+n_{2} n_{3}\right)$
$+2\left(l_{3} l_{1}+m_{3} m_{1}+n_{3} n_{1}\right) $
$= 3 $
$\Rightarrow \left(l_{1}+l_{2}+l_{3}\right)^{2}+\left(m_{1}+m_{2}+m_{3}\right)^{2}$
$+\left(n_{1}+n_{2}+n_{3}\right)^{2}=3$
Here, direction cosines of required line are
$\left(\frac{l_{1}+l_{2}+l_{3}}{\sqrt{3}}, \frac{m_{1}+m_{2}+m_{3}}{\sqrt{3}}, \frac{n_{1}+n_{2}+n_{3}}{\sqrt{3}}\right)$