The unit vectors orthogonal to (3 i +2 j +6 k ) and coplanar with (2 i + j + k ) and ( i - j + k ) are

Step-by-step solution

A vector which is coplanar to the vectors (2 i + j + k ) and ( i + j + k ) is ( a (2 i + j + k )+ b ( i - j + k ) ) ((2 a+b) i +(a-b) j +(a+b) k ) ...(i) Eq. (i) is orthogonal to (3 i +2 j +6 k ) ( aligned & 3(2 a+b)+2(a-b)+6(a+b) =0 & 14 a+7 b =0 & 2 a+b =0 aligned ) Required unit vector, ( aligned & = (2 a+b) i +(a-b) j +(a+b) k (2 a+b)^2+(a-b)^2+(a+b)^2 & = 3 a j -a k (3 a)^2+a^2 = 3 j - k 10 aligned ) Hence, option (b) is correct.