A Vector Calculus Computation of the Volume of a Parallelpiped

This is a temporary  pedagogical entry of a simple vector triple product calculation required in an application to a potential employer.

Problem:

Find the volume of the parallelepiped defined by the three vectors

A = (2, 1, 3) , \; B = (-5, 2, 1), \; C = (2, 1, 1)

by calculating the triple product A \cdot (B \times C) using the dot and cross product rules.

Solution:

Express the vectors in terms of the canonical orthonormal vector basis \hat{x}, \hat{y}, \hat{z} as

A = 2 \hat{x}+ 1 \hat{y}+ 3 \hat{z},

B = -5\hat{x}+ 2 \hat{y}+ 1\hat{z},

C = 2 \hat{x}+ 1 \hat{y}+ 1 \hat{z}.

Compute the vector cross product first:

P = B \times C = (-5 \hat{x} \times \hat{y}) + (-5 \hat{x} \times \hat{z}) + (2\hat{y}\times 2\hat{x}) + (2\hat{y} \times \hat{z})+(\hat{z} \times 2\hat{x})+(\hat{z} \times \hat{y})

= -5\hat{z}+(-5)(-\hat{y})+4(-\hat{z})+2\hat{x}+2\hat{y}+-\hat{x}

=\hat{x} +7\hat{y}-9\hat{z}= (1,7,-9).

Now calculate the inner or dot product

A \cdot (B \times C) = A \cdot P = (2, 1, 3) \cdot (1,7,-9) = 2+7-27=-18.

So the unitless volume of the parallelpiped is 18.

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