## 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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