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

Options

A. ` ( - 3 ) / ( 5 sqrt ( 2 ) ), ( 5 ) / ( 5sqrt2 ) , ( 4 ) / ( 5sqrt 2 ) `
B. ` ( 3 ) /( 5sqrt 2 ) , ( - 5 ) /(5sqrt 2 ) , ( 4 ) / ( 5sqrt2 ) `
C. ` ( 3 ) /( 5sqrt 2 ) , ( 5 ) / ( 5sqrt 2) , ( 4 ) / ( 5sqrt 2 ) `
D. None of these

Answer

A

Explanation

Vector normal to given planes are ` n_ 1 = hati - hatj + 2 hatk and n _ 2 = 3 hati + hatj + hatk ` .
So , their line of intersection is parallel to the vector
` n = n _ 1 xx n _ 2 = |{:( hati , hatj , hatk ) , ( 1, -1, 2 ) , ( 3, 1 , 1 ) :}| = - 3 hati +5 hatj + 4hatk `
` rArr hatn = ( - 3 ) / ( 5 sqrt 2 ) hati + ( 5 ) / ( 3 sqrt 2 ) hatj + ( 4 ) /( 5sqrt 2 ) hatk `
Hence, direction cosines of the line are ` ( - 3 ) / ( 5sqrt2 ) , ( 5 ) / ( 5sqrt2) , ( 4 ) / ( 5sqrt2 ) `