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The direction cosines of the line x-y+2z...

The direction cosines of the line `x-y+2z=5, 3x+y+z=6` are

A

`(-3)/(5sqrt(2)),5/(5sqrt(2)),4/(5sqrt(2))`

B

`3/(5sqrt(2)),(-5)/(5sqrt(2)),4/(5sqrt(2))`

C

`3/(5sqrt(2)),5/(5sqrt(2)),4/(5sqrt(2))`

D

none of these

Text Solution

Verified by Experts

The correct Answer is:
A
Vectors normal to given planes are `vecn_(1)=hati-hatj+2hatk` and `vecn_(2)=3hati+hatj+hatk`.
So their line of intersection is parallel to the vector
`vecn=vecn_(1)xxvecn_(2)=|(hati,hatj,hatk),(1,-1,2),(3,1,1)|=-3hati+5hatj+4hatk`
`impliesvecn=(-3)/(5sqrt(2))hati+5/(5sqrt(2))hatj+4/(5sqrt(2))hatk`
Hence, direction cosines of the line are `(-3)/(5sqrt(2)),5/(5sqrt(2)),4/(5sqrt(2))`
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