Volume of Parallelopiped via Vector Cross Product

Volume of a Parallelepiped via the Scalar Triple Product:

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Find the volume of the parallelepiped with adjacent edges AB, AC, and AD, where the points are A(4, -3, -2), B(2, 0, 5), C(-3, 2, 1), and D(1, 3, 2).

The volume is given by the scalar triple product: AB · (AC ´ AD). First we need the three vectors:

AB = [2 - 4]i + [0 - (-3)]j + [5 - (-2)]k = -2i + 3j + 7k.

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

First, 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)|

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

= | 4 + 57 - 189 | = 136 cubic units

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