The magnitude of a vector which is orthogonal to the vector i + j + k and is coplanar with the vectors i + j +2 k and i +2 j + k is

Step-by-step solution

Let v be a vector orthogonal to a = i + j + k , so v · a = 0, and coplanar with b = i + j + 2 k and c = i + 2 j + k . Since v is orthogonal to a and coplanar with b and c , it must be parallel to a × ( b × c ). First, calculate b × c : i (1 1 - 2 2) - j (1 1 - 2 1) + k (1 2 - 1 1) = -3 i + j + k Then evaluate the cross product a × ( b × c ): i (1 1 - 1 1) - j (1 1 - 1 (-3)) + k (1 1 - 1 (-3)) = 0 i - 4 j + 4 k Thus, v is proportional to -4 j + 4 k , which has magnitude (-4)^2 + 4^2 = 32 = 4 2 The correct option is