A vector which is orthogonal to the vector a = i + 2 j + 3 k and coplanar with the vectors b = 3 i + 2 j and c = 2 i + j + 3 k is

Step-by-step solution

Let the required vector be v . Since v is coplanar with b and c , and orthogonal to a , it is parallel to the vector triple product a ( b c ) . Using the vector triple product expansion: v = a ( b c ) = ( a c ) b - ( a b ) c Given vectors are: a = i + 2 j + 3 k b = 3 i + 2 j c = 2 i + j + 3 k Calculating the dot products: a c = (1)(2) + (2)(1) + (3)(3) = 2 + 2 + 9 = 13 a b = (1)(3) + (2)(2) + (3)(0) = 3 + 4 + 0 = 7 Substituting these values into the expansion: v = 13(3 i + 2 j ) - 7(2 i + j + 3 k ) v = (39 i + 26 j