The volume of the tetrahedron whose coterminus edges are represented by a =-12 i + p k , b =3 j - k , c =2 i + j -15 k , is 570 cu. units, then p =

Step-by-step solution

The volume of a tetrahedron with coterminous edges a , b , c is given by V = 1 6 |[ a b c ]| , where [ a b c ] is the scalar triple product. Given vectors: a = (-12, 0, p) , b = (0, 3, -1) , c = (2, 1, -15) , and V = 570 . Compute the scalar triple product: [ a b c ] = vmatrix -12 & 0 & p 0 & 3 & -1 2 & 1 & -15 vmatrix = -12 vmatrix 3 & -1 1 & -15 vmatrix + p vmatrix 0 & 3 2 & 1 vmatrix = -12 [3 (-15) - (-1) 1] + p [0 1 - 3 2] = 528 - 6p . Substitute into the volume formula: 570 = 1 6 |528 - 6p| |528 - 6p| = 3420 T