If the volume of the tetrahedron whose coterminous edges are given by the vectors a = -2 i + 3 j - 3 k , b = 4 i + 5 j + ( - 10) k , c = 6 i + 2 j - 3 k is 11 cubic units, then the sum of the possible values of is

Step-by-step solution

The volume of a tetrahedron with coterminous edges a , b , c is given by V = 1 6 |[ a , b , c ]| . Given V = 11 , we have |[ a , b , c ]| = 66 . Calculating the scalar triple product: [ a , b , c ] = vmatrix -2 & 3 & -3 4 & 5 & - 10 6 & 2 & -3 vmatrix = -2(-15 - 2( - 10)) - 3(-12 - 6( - 10)) - 3(8 - 30) = -2(5 - 2 ) - 3(48 - 6 ) - 3(-22) = -10 + 4 - 144 + 18 + 66 = 22 - 88 Substituting this into the volume equation: |22 - 88| = 66 22| - 4| = 66 | - 4| = 3 This gives - 4 = 3 = 7 and - 4 = -3 = 1 . The possible value