lf $(l_{1}, m_{1}, n_{1})$ , $(l_{2}, m_{2}, n_{2})$ , $(l_{3}, m_{3}, n_{3})$ are the d.c.s of three mutually perpendicular lines, then the direction cosines of the line whose direction ratios are $l_{1} + l_{2} + l_{3}, m_{1} + m_{2} + m_{3}, n_{1} + n_{2} + n_{3}$ are

$\frac{l_{1} + l_{2} + l_{3}}{3}, \frac{m_{1} + m_{2} + m_{3}}{3}, \frac{n_{1} + n_{2} + n_{3}}{3}$

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$\frac{l_{1} + l_{2} + l_{3}}{\sqrt{2}}, \frac{m_{1} + k + m_{3}}{\sqrt{2}}, \frac{n_{1} + n_{2} + n_{3}}{\sqrt{2}}$

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l_{1} l_{2} l_{3}, m_{1} m_{2} m_{3}, n_{1} n_{2} n_{3}

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$\frac{l_{I} + l_{2} + l_{3}}{\sqrt{3}}, \frac{m_{1} + n + n}{\sqrt{3}}, \frac{n_{I} + n_{2} + n_{3}}{\sqrt{3}}$

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Solution

The correct option is D
$\frac{l_{I} + l_{2} + l_{3}}{\sqrt{3}}, \frac{m_{1} + n + n}{\sqrt{3}}, \frac{n_{I} + n_{2} + n_{3}}{\sqrt{3}}$

Dr of line is ( $L_{1} + L_{2} + L_{3}$ , $M_{1} + M_{2} + M_{3}$ , $N_{1} + N_{2} + N_{3}$ )
Where the given three lines are perpendicular to each other.
For finding the Dc of required line we have to divide by magnitude
$\sqrt{{(L_{1} + L_{2} + L_{3})}^{2} + {(M_{1} + M_{2} + M_{3})}^{2} + {(N_{1} + N_{2} + N_{3})}^{2}}$
On expanding, we have
$\sqrt{(l_{1})^{2} + (l_{2})^{2} + (l_{3})^{2} + 2 (l_{1} l_{2} + l_{2} l_{3} + l_{1} l_{3}) + (m_{1})^{2} + (m_{2})^{2} + (m_{3})^{2} + 2 (m_{1} m_{2} + m_{2} m_{3} + m_{1} m_{3}) + (l_{1})^{2} + (n_{2})^{2} + (n_{3})^{2} + 2 (n_{1} n_{2} + n_{2} n_{3} + n_{1} n_{3})}$
Now use $(l_{1})^{2} + (m_{1})^{2} + (n_{1})^{2}$ $= 1$
$(l_{2})^{2} + (m_{2})^{2} + (n_{2})^{2}$ $= 1$
$(l_{3})^{2} + (m_{3})^{2} + (n_{3})^{2}$ $= 1$
$l_{1} l_{2} + m_{1} m_{2} + n_{1} n_{2}$ $= 0$
$l_{2} l_{3} + m_{2} m_{3} + n_{2} n_{3}$ $= 0$
$l_{1} l_{3} + m_{1} m_{3} + n_{1} n_{3}$ $= 0$ using perpendicular condition
So DC of line will be:
$(\frac{L_{1} + L_{2} + L_{3}}{\sqrt{3}}, \frac{M_{1} + M_{2} + M_{3}}{\sqrt{3}}, \frac{N_{1} + N_{2} + N_{3}}{\sqrt{3}})$

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Similar questions

If $l_{1}, m_{1}, n_{1}, l_{2}, m_{2}, n_{2} a n d l_{3}, m_{3}, n_{3}$ are the direction cosines of three mutually perpendicular lines,then prove that the line whose direction cosines are proportional to $l_{1} + l_{2} + l_{3}, m_{1} + m_{2} + m_{3} a n d n_{1} + n_{2} + n_{3}$ makes equal angles with them.

If three mutually perpendicular lines have direction cosines $(l_{1}, m_{1}, n_{1}), (l_{2}, m_{2}, n_{2})$ and $(l_{3}, m_{3}, n_{3})$ line having direction cosines $l_{1} + l_{2} + l_{3}, m_{1} + m_{2} + m_{3}$ and $n_{1} + n_{2} + n_{3}$ make an
angle of..... with each other