The Volume of a Parallelepiped

Section 1.6 The Volume of a Parallelepiped

We now want to derive a formula to compute the volume of a parallelepiped spanned by three vectors in \(\mathbb R^3\text{.}\)

Consider the parallelepiped spanned by the vectors \(\vect x,\vect y,\vect z\in\mathbb R^3\) as shown in Figure 1.28.

Figure 1.28. A parallelepiped spanned by three vectors.

The next theorem establishes a formula for the volume.

Theorem 1.29. Volume of parallelepiped.

The volume, \(V\text{,}\) of the parallelepiped spanned by the three vectors \(\vect x=(x_1,x_2,x_3)\text{,}\) \(\vect y=(y_1,y_2,y_3)\) and \(\vect z=(z_1,z_2,z_3)\) is given by

\begin{equation} V =\left|\det \begin{bmatrix} x_1\amp y_1\amp z_1\\ x_2\amp y_2\amp z_2\\ x_3\amp y_3\amp z_3 \end{bmatrix} \right|\text{.}\tag{1.13} \end{equation}

Proof.

The volume of the parallelepiped is the area of its base times its height. The height is \(\|\vect z\|\lvert\cos(\theta)\rvert\text{,}\) where \(\theta\) is the angle between \(\vect z\) and the vector \(\vect x\times\vect y\) perpendicular to the plane spanned by \(\vect x\) and \(\vect y\text{.}\) By (1.2)

\begin{equation*} \|\vect z\| |\cos(\theta)| =\|\vect z\|\frac{|\vect z\cdot(\vect x\times\vect y)|} {\|\vect z\|\|\vect x\times\vect y\|} =\frac{|\vect z\cdot(\vect x\times\vect y)|} {\|\vect x\times\vect y\|}\text{.} \end{equation*}

According to Theorem 1.24 the surface area of the parallelogram spanned by \(\vect x\) and \(\vect y\) given by \(\|\vect x\times\vect y\|\text{.}\) Hence we deduce that the volume of the parallelepiped is

\begin{equation*} V=\frac{|\vect z\cdot(\vect x\times\vect y)|} {\|\vect x\times\vect y\|}\|\vect x\times\vect y\| =|\vect z\cdot(\vect x\times\vect y)|\text{.} \end{equation*}

Using Lemma 1.22 we see that

\begin{equation*} \vect z\cdot(\vect x\times\vect y) =\det \begin{bmatrix} x_1\amp y_1\amp z_1\\ x_2\amp y_2\amp z_2\\ x_3\amp y_3\amp z_3 \end{bmatrix}\text{,} \end{equation*}

from which (1.13) follows.

Remark 1.30.

Similar as in case of a \(2\times 2\) matrix this provides a geometrical interpretation of the determinant of a \(3\times 3\) matrix. It is, up to a sign, the volume of the parallelepiped spanned by the columns of the matrix. The sign indicates whether the triple of vectors given by the colums of the matrix is positively or negatively oriented.

What we discussed above generalises to higher dimensions.

Remark 1.31. Volume in higher dimensions.

One can also define a “volume” or “measure” for subsets of \(\mathbb R^N\) if \(N>3\text{.}\) One starts with “rectangular boxes,” and defines their volume to be the product of its \(N\) sides as one does in the plane and in space. Then one can look at parallelepipeds. It turns out that the “volume” of an \(N\)-dimensional parallelepiped spanned by the \(N\) (linearly independent) vectors \(\vect v_1,\dots,\vect v_N\) is given by the absolute value of \(\det[\vect v_1 \dots \vect v_N]\) as in case of \(N=2\) or \(N=3\text{.}\)

More generally we can compute the “\(k\)-dimensional” volume of \(k\) vectors \(\vect v_1,\dots,\vect v_k\) in \(\mathbb R^N\text{,}\) generalising the formula given in Theorem 1.29. We form the matrix \(J\) having columns \(\vect v_1,\dots,\vect v_k\text{.}\) It turns out that the volume required is

\begin{equation*} V=\sqrt{\det(J^TJ)}\text{.} \end{equation*}

For an elementary proof of this fact see for instance Section 5.3 of [3].

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