The direction cosines of three mutually perpendicular straight lines are l1,m1,n1,l2,m2,n2,l3,m3,n3. Then the direction cosines of a line which is equally inclined to the given three lines, are

# The direction cosines of three mutually perpendicular straight lines are $〈{l}_{1},{m}_{1},{n}_{1}〉,〈{l}_{2},{m}_{2},{n}_{2}〉,〈{l}_{3},{m}_{3},{n}_{3}〉$. Then the direction cosines of a line which is equally inclined to the given three lines, are

1. A

$\left({l}_{1}+{l}_{2}+{l}_{3}\right),\left({m}_{1}+{m}_{2}+{m}_{3}\right),\left({n}_{1}+{n}_{2}+{n}_{3}\right)$

2. B

$\left(\frac{{l}_{1}+{l}_{2}+{l}_{3}}{3}\right),\left(\frac{{m}_{1}+{m}_{2}+{m}_{3}}{3}\right),\left(\frac{{n}_{1}+{n}_{2}+{n}_{3}}{3}\right)$

3. C

$\left(\frac{{l}_{1}+{l}_{2}+{l}_{3}}{\sqrt{3}}\right),\left(\frac{{m}_{1}+{m}_{2}+{m}_{3}}{\sqrt{3}}\right),\left(\frac{{n}_{1}+{n}_{2}+{n}_{3}}{\sqrt{3}}\right)$

4. D

$\left(\frac{{l}_{1}+{l}_{2}+{l}_{3}}{\sqrt{2}}\right),\left(\frac{{m}_{1}+{m}_{2}+{m}_{3}}{\sqrt{2}}\right),\left(\frac{{n}_{1}+{n}_{2}+{n}_{3}}{\sqrt{2}}\right)$

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### Solution:

Given  $〈{l}_{1},{m}_{1},{n}_{1}〉,〈{l}_{2},{m}_{2},{n}_{2}〉,〈{l}_{3},{m}_{3},{n}_{3}〉$are the direction cosines of three mutually perpendicular lines

It implies that $\sum {l}_{1}{l}_{2}=\sum {l}_{2}{l}_{3}=\sum {l}_{1}{l}_{3}=0,\sum {{l}_{1}}^{2}=\sum {{l}_{2}}^{2}=\sum {{l}_{3}}^{2}=1$

Hence the direction cosines of line which is equally inclined to the above lines are $\left(\frac{{l}_{1}+{l}_{2}+{l}_{3}}{3},\frac{{m}_{1}+{m}_{2}+{m}_{3}}{3},\frac{{n}_{1}+{n}_{2}+{n}_{3}}{3},\right)$

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