The unit vector which is orthogonal to the vector `3hati+2hatj+6hatk` and is coplanar with the vectors `2hati+hatj+hatk` and `hati -hatj+hatk` is

Options

A. `(2hat(i) -6hat(j)+hat(k))/(sqrt(41))`
B. `(2hat(i) - 3hat(j))/(sqrt(13))`
C. `(3hat(j) - hat(k))/(sqrt(10))`
D. `(4hat(i) +3hat(j)-3hat(k))/(sqrt(34))`

Answer

C

Explanation

As we know that a vector coplanar to `vec(a) ". " vec(b)` and orthogonal to `vec(c ) " is " lambda {(vec(a) ". " vec(b)) xx vec(c )}`
`:.` A vector coplanar to `(2hat(i) + hat(j) + hat(k)) , (hat(i) - hat(j) + hat(k)) ` and orthogonal to `3hat(i) + 2hat(j) +6hat(k)`
`= lambda [{(2hat(i) + hat(j) + hat(k)) xx (hat(i) - hat(j)+ hat(k))} xx (3hat(i) + 2hat(j) + 6hat(k))]`
`= lambda [(2hat(i) -hat(j) -3hat(k)) xx (3hat(i) + 2hat(j) +6hat(k))]`
`=lambda (21 hat(j) = 7hat(k))`
`:.` Unit vectors `=+ ((21 hat(j) -7hat(k)))/(sqrt((21)^(2) (7)^(2))) =+ ((3hat(j) -hat(k)))/( sqrt(10))`