vectors problem

Question

Find the vector and scalar equations for the plane through A(1, 0, 10), B(5, -5, 3), and C(8, 8, 2).

Answer 1

AB = <4, - 5, - 7>; AC = <7, 8, - 8>; r = <1, 0, 10> + s<4, - 5, - 7> + t<7, 8, - 8>, s,t are any real numbers,

or r = <1 + 4s + t, - 5s + 8t, 10 - 7s - 8t>

AB x AC = det [(i j k) (4 -5 -7) (7 8 -8)] = 96i-17j + 67k is normal vector to plane.

So, scalar equation of plane: 96(x - 1) - 17y + 67(z - 10) = 0