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The direction cosines of a line equally ...

The direction cosines of a line equally inclined to three mutually perpendiclar 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

`(l_(1)+l_(2)+l_(3))/sqrt(3), (m_(1)+m_(2)+m_(3))/sqrt(3), (n_(1)+n_(2)+n_(3))/sqrt(3)`

C

`(l_(1)+l_(2)+l_(3))/3 ,(m_(1)+m_(2)+m_(3))/(3), (n_(1)+n_(2)+n_(3))/(3)`

D

none of these

Text Solution

Verified by Experts

The correct Answer is:
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)=1, l_(2)^(2)+m_(2)^(2)+n_(2)^(2)=1.`
`l_(3)^(2)+m_(3)^(2)+n_(3)^(2)=1`
Now, `(l_(1)+l_(2)+l_(3))^(2)+(m_(1)+m_(2)+m_(3))^(2)+(n_(1)+n_(2)+n_(3))^(2)`
`=(l_(1)^(2)+m_(1)^(2)+n_(1)^(2)+(l_(2)^(2)+m_(2)^(2)+n_(2)^(2))+(l_(3)^(2)+m_(3)^(2)+n_(3)^(2))+2(l_(1)l_(2)+m_(1)m_(2)+n_(1)n_(2))+2(l_(2)l_(3)+m_(2)m_(3)+n_(2)n_(3))+2(l_(3)l_(1)+m_(3)m_(1)+n_(3)n_(1))=3`
`rArr (l_(1)+l_(2)+_(3))^(2) + (m_(1)+m_(2)+m_(3))^(2) + (n_(1)+n_(2)+n_(3))^(2)+(m_(1)m_(2)+m_(3))^(2)+(n_(1)+n_(2)+n_(3))^(2)=3`
Hence, direction cosines of required line are
`(l_(1)+l_(2)+l_(3))/sqrt(3), (m_(1)+m_(2)+m_(3))/sqrt(3), (n_(1)+n_(2)+n_(3))/sqrt(3)`
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