The sum of all real values of for which the vectors a = i + j + k , b = i + j + 2 k , c = 2 i + 3 j + k are coplanar is...

Step-by-step solution

For the vectors a , b , and c to be coplanar, their scalar triple product must be zero. vmatrix & 1 & 1 1 & & 2 2 & 3 & vmatrix = 0 Expanding the determinant: ( ^2 - 6) - 1( - 4) + 1(3 - 2 ) = 0 ^3 - 6 - + 4 + 3 - 2 = 0 ^3 - 9 + 7 = 0 Let f( ) = ^3 - 9 + 7 . f'( ) = 3 ^2 - 9 = 0 = 3 f( 3 ) = 7 - 6 3 0 Since the local maximum is positive and the local minimum is negative, the cubic equation has three distinct real roots. The sum of the roots of the cubic equation A ^3 + B ^2 + C + D = 0 is given by - B A . Here, the