The volume of a tetrahedron with vertices 5 i - j + k , 7 i - 4 j + p k , i - 6 j + 10 k and - i - 3 j + 7 k is 11 cubic units, then one of the values of p is

Step-by-step solution

Let the vertices of the tetrahedron be A(5, -1, 1) , B(7, -4, p) , C(1, -6, 10) and D(-1, -3, 7) . The position vectors of the edges from vertex A are: AB = 2 i - 3 j + (p - 1) k AC = -4 i - 5 j + 9 k AD = -6 i - 2 j + 6 k The volume of the tetrahedron is given by V = 1 6 |[ AB , AC , AD ]| . [ AB , AC , AD ] = vmatrix 2 & -3 & p - 1 -4 & -5 & 9 -6 & -2 & 6 vmatrix = 2(-30 + 18) + 3(-24 + 54) + (p - 1)(8 - 30) = 2(-12) + 3(30) - 22(p - 1) = -24 + 90 - 22p + 22 = 88 - 22p Given V = 11 , we have: 1 6 |88 - 22p| = 11