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How to Calculate Volume of Prism and Pyramid with Vectors

You can use the cross product and dot product to find the volume of a square prism (parallelepiped), a pyramid, or a tetrahedron (a pyramid with a triangular base) that is spanned by three vectors. To do this, you first find the cross product of two of the vectors, next the dot product of that cross product and the third vector, and finally multiply the number you get by a number belonging to the figure you want to find the volume of.

For a square prism, you don’t need to multiply by anything. For a pyramid with a square base, the factor is $\frac{1}{3}$ , and for a tetrahedron, the factor is $\frac{1}{6}$ . Note! Sometimes you get a negative number. A negative volume makes no sense, so you should always use the absolute value of your answer, for example $V = | - 14 | = 14$ .

Formula

The Volume of a Square Prism

V_{F} = | (\vec{u} \times \vec{v}) \cdot \vec{w} |

A prism spanned by three vectors

Example 1

Find the volume of the square prism spanned by $(1, 3, - 2)$ , $(- 3, 2, 4)$ and $(1, 1, 1)$ .

You now have to find the cross product of two of the vectors, and then find the dot product of this cross product and the unused vector. The cross product of the first two vectors is $(16, 2, 11)$ . Then you find the dot product of that and $(1, 1, 1)$ ,

(16, 2, 11) \cdot (1, 1, 1) = 16 + 2 + 11 = 29,

which gives us a volume of 29. If you had gotten a negative answer, you would have had to use the absolute value of that as the volume.

Formula

The Volume of a Pyramid

V_{P} = \frac{1}{3} | (\vec{u} \times \vec{v}) \cdot \vec{w} |

Note! This formula is only applicable when the base of the pyramid is a square. When the base is a triangle, the figure is called a tetrahedron, which has its own formula.

Volume of the pyramid spanned by the three vectors

Example 2

Find the volume of a pyramid that is spanned by $(1, 3, - 2)$ , $(- 3, 2, 4)$ and $(1, 1, 1)$ .

You begin by finding the cross product of two of the vectors, and then you find the dot product of that cross product and the unused vector. Then you multiply the answer by $\frac{1}{3}$ . The cross product of the two first vectors is $(16, 2, 11)$ . The dot product of this and $(1, 1, 1)$ is $29$ , which you multiply with $\frac{1}{3}$ :

\frac{1}{3} \cdot 29 = \frac{29}{3}

The volume is $\frac{29}{3}$ . If you had gotten a negative answer, you would have had to take the absolute value of it as the volume.

Formula

The Volume of a Tetrahedron

V_{T} = \frac{1}{6} | (\vec{u} \times \vec{v}) \cdot \vec{w} |

The volume of a tetrahedron spanned by three vectors

Example 3

Find the volume of a tetrahedron that is spanned by $(1, 3, - 2)$ , $(- 3, 2, 4)$ and $(1, 1, 1)$ .

You begin by finding the cross product of two of the vectors, and then find the dot product of that cross product and the unused vector. Then you multiply the answer with $\frac{1}{6}$ . The cross product of the first two vectors is $(16, 2, 11)$ . The dot product of this and $(1, 1, 1)$ is $29$ , which you multiply with $\frac{1}{6}$ :

\frac{1}{6} \cdot 29 = \frac{29}{6}

The volume is $\frac{29}{6}$ . If you had gotten a negative answer, you would have had to take the absolute value of it as the volume.

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