Let a , ~b , c , d are vectors such that a b =2 i +3 j - k and c d =3 i +2 j + k and if | array cc a c & b c a ~d & ~b ~d array |=0 , then =

Step-by-step solution

A fundamental identity in vector algebra relates mixed scalar and vector products: | array cc a c & b c a d & b d array | = ( a b ) ( c d ) Given that this determinant equals zero, the dot product of the cross products must also vanish: ( a b ) ( c d ) = 0 Substituting the provided expressions a b = 2 i + 3 j - k and c d = 3 i + 2 j + k yields: (2)(3) + (3)(2) + (-1)( ) = 0 6 + 6 - = 0 = 12 The correct choice is option C.