Multivariable calculus and vector analysis

The relationship between determinants and area or volume

From the properties of the cross product and the scalar triple product, we can discover a link between $2\times 2$ determinants and area, and a link between $3\times 3$ determinants and volume.

2 $\times$ 2 determinants and area

Recall that the area of the parallelogram spanned by $\mathbf{a}$ and $\mathbf{b}$ is the magnitude of $\mathbf{a}\times \mathbf{b}$. We can write the cross product of $\mathbf{a}=a_1\mathbf{i}+a_2\mathbf{j}+a_3\mathbf{k}$ and $\mathbf{b}=b_1\mathbf{i}+b_2\mathbf{j}+b_3\mathbf{k}$ as the determinant

$$\begin{array}{lll}\hfill \mathbf{a}\times \mathbf{b}=\left|\begin{array}{ccc}\hfill \mathbf{i}\hfill & \hfill \mathbf{j}\hfill & \hfill \mathbf{k}\hfill \\ \hfill a_1\hfill & \hfill a_2\hfill & \hfill a_3\hfill \\ \hfill b_1\hfill & \hfill b_2\hfill & \hfill b_3\hfill \end{array}\right|.& & \hfill \end{array}$$

Now, imagine that $\mathbf{a}$ and $\mathbf{b}$ lie in the plane so that $a_3=b_3=0$. Using our rules for calculating determinants, we see that, in this case, the cross product simplifies to

$$\begin{array}{lll}\hfill \mathbf{a}\times \mathbf{b}=\left|\begin{array}{cc}\hfill a_1\hfill & \hfill a_2\hfill \\ \hfill b_1\hfill & \hfill b_2\hfill \end{array}\right|\mathbf{k}.& & \hfill \end{array}$$

Hence, the area of the parallelogram, $\left|\right|\mathbf{a}\times \mathbf{b}\left|\right|$, is the absolute value of the determinant

$$\begin{array}{lll}\hfill \left|\begin{array}{cc}\hfill a_1\hfill & \hfill a_2\hfill \\ \hfill b_1\hfill & \hfill b_2\hfill \end{array}\right|.& & \hfill \end{array}$$

As mentioned in our discussion about determinant notation, it’s difficult to represent the absolute value of a determinant using the above notation. Instead, we write that the area of the parallelogram spanned by $\mathbf{a}=a_1\mathbf{i}+a_2\mathbf{j}$ and $\mathbf{b}=b_1\mathbf{i}+b_2\mathbf{j}$ is

$$\begin{array}{lll}\hfill \left|\right|\mathbf{a}\times \mathbf{b}\left|\right|=\left|det\left(\left[\begin{array}{cc}\hfill a_1\hfill & \hfill a_2\hfill \\ \hfill b_1\hfill & \hfill b_2\hfill \end{array}\right]\right)\right|.& & \hfill \end{array}$$

3 $\times$ 3 determinants and volume

The volume of a parallelepiped spanned by the vectors $\mathbf{a}$, $\mathbf{b}$ and $\mathbf{c}$ is the absolute value of the scalar triple product $\left(\mathbf{a}\times \mathbf{b}\right)\cdot \mathbf{c}$. We can write the scalar triple product of $\mathbf{a}=a_1\mathbf{i}+a_2\mathbf{j}+a_3\mathbf{k}$, $\mathbf{b}=b_1\mathbf{i}+b_2\mathbf{j}+b_3\mathbf{k}$, and $\mathbf{c}=c_1\mathbf{i}+c_2\mathbf{j}+c_3\mathbf{k}$ as the determinant

$$\begin{array}{lll}\hfill \left(\mathbf{a}\times \mathbf{b}\right)\cdot \mathbf{c}=\left|\begin{array}{ccc}\hfill a_1\hfill & \hfill a_2\hfill & \hfill a_3\hfill \\ \hfill b_1\hfill & \hfill b_2\hfill & \hfill b_3\hfill \\ \hfill c_1\hfill & \hfill c_2\hfill & \hfill c_3\hfill \end{array}\right|.& & \hfill \end{array}$$

Hence, the volume of the parallelepiped spanned by $\mathbf{a}$, $\mathbf{b}$, and $\mathbf{c}$ is

$$\begin{array}{lll}\hfill |\left(\mathbf{a}\times \mathbf{b}\right)\cdot \mathbf{c}|=\left|det\left(\left[\begin{array}{ccc}\hfill a_1\hfill & \hfill a_2\hfill & \hfill a_3\hfill \\ \hfill b_1\hfill & \hfill b_2\hfill & \hfill b_3\hfill \\ \hfill c_1\hfill & \hfill c_2\hfill & \hfill c_3\hfill \end{array}\right]\right)\right|.& & \hfill \end{array}$$

Contents by topic

1. Geometry in higher dimensions

1a. Preliminaries

1b. Lines and planes

1c. Matrices

• Matrices and determinants
• Multiplying matrices and vectors
• Dot product in matrix notation
• Matrices and linear functions
• The relationship between determinants and area or volume
• Matrix and vector multiplication examples

1d. Visualizing functions in higher dimensions

2. Differential Calculus

3. Integral calculus

4. Vector calculus

5. The fundamental theorems of vector calculus

6. Review material