4 Points Coplanar via Scalar Triple Product: Show that the four points, A, B, C, and D, are coplanar:

A(4, -3, -2), B(11, -8, -5), C(-3, 2, 1), and D(1, 3, 2).

The scalar triple product will give us the volume of the parallelepiped formed by the vectors AB, AC, and AD. If this volume is zero, it indicates that the points are coplanar.

First we need the three vectors:

AB = [11 - 4]i + [-8 - (-3)]j + [-5 - (-2)]k = 7i - 5j - 3k.

AC = [-3 - 4]j + [2 - (-3)]j + [1 - (-2)]k = -7i + 5j + 3k

AD = [1 - 4]i + [3 - (-3)]j + [2 - (-2)]k = -3i + 6j + 4k

Next, find the cross product:

= i[20 - 18] - j[-28 - (-9)] + k[-42 - (-15)] = 2i + 19j - 27k

Now form the dot product to get the volume:

Volume = |AB · (2i + 19j - 27k)|

= | (7i - 5j - 3k) · (2i + 19j - 27k)|

= | 14 - 95 + 81 | = 0.

Hence the points are coplanar.