Let ` vec a= hat i- hat j ,\ vec b=3 hat j- hat k` and `vec c=7 hat i- hat kdot\ `. Find a vector ` vec d` which is perpendicular to both ` vec a` and `vec b` , and `vecc cdot vec d=1.`

Options

A. `vec(d)= 1/4(hat(i) - hat(j) + 3 hat(k))`
B. `vec(d)= hat(i) + hat(j) + 3 hat(k)`
C. `vec(d)= 1/4(hat(i) + hat(j) + 3 hat(k))`
D. `vec(d)= 1/4(hat(i) + hat(j) + hat(k))`

Answer

C

Explanation

Since `vec(d)` is perpendicular to both `vec(a) and vec(b)` , so it is parallel to ` ( vec(a) xx vec(b))`
`:. vec(d) = lambda (vec(a) xx vec(b)) = lambda ( hat(i) + hat(j)+ 3 hat(k) ) = ( lambda hat(i) + lambda hat(j) + 3 lambda hat(k))`
`vec(c)*vec(d) = 1 rArr ( 7 hat(i) - hat(k))* (lambdahat(i) + lambda hat(j)+ 3 lambda hat(k)) = 1`
`rArr 7 lambda - 3 lambda =1 rArr4 lambda = 1 rArr lambda =1/4.`
`:. vec(d) = 1/4 ( hat(i)+ hat(j)+3 hat(k)).`