Finding the Volume of a Parallelepiped

A parallelepiped is a three-dimensional geometric shape with six rectangular faces. To find the volume of a parallelepiped, we can use the formula:

Volume = |a · (b × c)|

where a, b, and c are the three adjacent edge vectors of the parallelepiped.

In this problem, we are given the coordinates of four points that define the parallelepiped: P(2, 0, 2), Q(-2, 3, 6), R(5, 3, 0), and S(-1, 6, 4).

To find the volume, we first need to calculate the three adjacent edge vectors:

1. PQ = Q - P = (-2, 3, 6) - (2, 0, 2) = (-4, 3, 4)
2. PR = R - P = (5, 3, 0) - (2, 0, 2) = (3, 3, -2)
3. PS = S - P = (-1, 6, 4) - (2, 0, 2) = (-3, 6, 2)

Now, we can calculate the cross product of two of the edge vectors to get the third vector:

b × c = (3, 3, -2) × (3, 6, 2) = (0, -4, 18)

Finally, we can calculate the volume using the formula:

Volume = |a · (b × c)| = |(3, 3, 4) · (0, -4, 18)| = |54| = 54 units³

Therefore, the volume of the parallelepiped with the given vertices is 54 units³.