A unit vector coplanar with i + j + 2 k and i + 2 j + k and perpendicular to i + j + k is

Step-by-step solution

Let the required vector be v = x i + y j + z k . Since v is coplanar with a = i + j + 2 k and b = i + 2 j + k , it must be perpendicular to their cross product a b . a b = vmatrix i & j & k 1 & 1 & 2 1 & 2 & 1 vmatrix = -3 i + j + k Thus, v ( a b ) = 0 -3x + y + z = 0 Also, v is perpendicular to c = i + j + k , so v c = 0 . x + y + z = 0 Subtracting the two equations gives -4x = 0 x = 0 . Substituting x = 0 into x + y + z = 0 gives y = -z . So the vector is of the form v = y j - y k = y( j - k ) . Since v is a unit