The volume of the tetrahedron whose vertices are A (-1, 2, 3) , B (3, -2, 1) , C (p, 1, 3) , D (-1, -2, 4) is 16 3 cubic units then the value of p is

Step-by-step solution

The volume of a tetrahedron with vertices A, B, C, D is given by V = 1 6 |[ AB , AC , AD ]| . The vectors representing the edges from A are: AB = (3 - (-1)) i + (-2 - 2) j + (1 - 3) k = 4 i - 4 j - 2 k AC = (p - (-1)) i + (1 - 2) j + (3 - 3) k = (p+1) i - j AD = (-1 - (-1)) i + (-2 - 2) j + (4 - 3) k = -4 j + k The scalar triple product is: [ AB , AC , AD ] = vmatrix 4 & -4 & -2 p+1 & -1 & 0 0 & -4 & 1 vmatrix Expanding the determinant: = 4(-1 - 0) + 4(p+1 - 0) - 2(-4(p+1) - 0) = -4 + 4p + 4 + 8p + 8 = 12p + 8 Give