Multivariable calculus and vector analysis

The component formula for the cross product

The geometric definition of the cross product is a good for understanding the properties of the cross product. However, the geometric definition isn’t so useful for computing the cross product of vectors. For computations, we will want a formula in terms of the components of vectors. We start by using the geometric definition to compute the cross product of the standard unit vectors.

Cross product of unit vectors

Let $\mathbf{i}$, $\mathbf{j}$, and $\mathbf{k}$ be the standard unit vectors in ${\mathbf{R}}^3$. (We define the cross product only in three dimensions.)

The parallelogram spanned by any two of these standard unit vectors is a unit square, which has area one. Hence, by the geometric definition, the cross product must be a unit vector. Since the cross product must be perpendicular to the two unit vectors, it must be equal to the other unit vector or the opposite of that unit vector. Looking at the above graph, you can use the right-hand rule to determine the following results.

$$\begin{array}{llll}\hfill \mathbf{i}\times \mathbf{j}& =\mathbf{k}& \hfill & \\ \hfill \mathbf{j}\times \mathbf{k}& =\mathbf{i}& \hfill & \\ \hfill \mathbf{k}\times \mathbf{i}& =\mathbf{j}& \hfill & \end{array}$$

This little cycle diagram can help you remember these results.

What about $\mathbf{i}\times \mathbf{k}$? By the right-hand rule, it must be $-\mathbf{j}$. I think you get the idea (after all $\mathbf{b}\times \mathbf{a}=-\mathbf{a}\times \mathbf{b}$).

$$\begin{array}{llll}\hfill \mathbf{j}\times \mathbf{i}& =-\mathbf{k}& \hfill & \\ \hfill \mathbf{k}\times \mathbf{j}& =-\mathbf{i}& \hfill & \\ \hfill \mathbf{i}\times \mathbf{k}& =-\mathbf{j}& \hfill & \end{array}$$

Finally, the cross product of any vector with itself is the zero vector ($\mathbf{a}\times \mathbf{a}=\mathbf{0}$). So the cross product of any standard unit vector with itself the zero vector.

General vectors

With the exception of the two special properties mentioned above ($\mathbf{b}\times \mathbf{a}=-\mathbf{a}\times \mathbf{b}$, and $\mathbf{a}\times \mathbf{a}=\mathbf{0}$), the cross product behaves like regular multiplication. It obeys the following properties:

• $\left(y\mathbf{a}\right)\times \mathbf{b}=y\left(\mathbf{a}\times \mathbf{b}\right)=\mathbf{a}\times \left(y\mathbf{b}\right)$,
• $\mathbf{a}\times \left(\mathbf{b}+\mathbf{c}\right)=\mathbf{a}\times \mathbf{b}+\mathbf{a}\times \mathbf{c}$,
• $\left(\mathbf{b}+\mathbf{c}\right)\times \mathbf{a}=\mathbf{b}\times \mathbf{a}+\mathbf{c}\times \mathbf{a}$,

where $\mathbf{a}$, $\mathbf{b}$, and $\mathbf{c}$ are vectors in $R^3$ and $y$ is a scalar. We can use these properties, along with the cross product of the standard unit vectors, to write the formula for the cross product in terms of components.

We write the components of $\mathbf{a}$ and $\mathbf{b}$ as:

$$\begin{array}{lll}\hfill \mathbf{a}=\left(a_1,a_2,a_3\right)=a_1\mathbf{i}+a_2\mathbf{j}+a_3\mathbf{k}& & \hfill \\ \hfill \mathbf{b}=\left(b_1,b_2,b_3\right)=b_1\mathbf{i}+b_2\mathbf{j}+b_3\mathbf{k}& & \hfill \end{array}$$

First, we’ll assume that $a_3=b_3=0$. (Then, the manipulations are much easier.) We calculate:

$$\begin{array}{llll}\hfill \mathbf{a}\times \mathbf{b}& =\left(a_1\mathbf{i}+a_2\mathbf{j}\right)\times \left(b_1\mathbf{i}+b_2\mathbf{j}\right)& \hfill & \\ \hfill & =a_1{b}_1\left(\mathbf{i}\times \mathbf{i}\right)+a_1{b}_2\left(\mathbf{i}\times \mathbf{j}\right)+a_2{b}_1\left(\mathbf{j}\times \mathbf{i}\right)+a_2{b}_2\left(\mathbf{j}\times \mathbf{j}\right)& \hfill & \end{array}$$

Since we know that $\mathbf{i}\times \mathbf{i}=\mathbf{0}=\mathbf{j}\times \mathbf{j}$ and that $\mathbf{i}\times \mathbf{j}=\mathbf{k}=-\mathbf{j}\times \mathbf{i}$, this quickly simplifies to

$$\begin{array}{llll}\hfill \mathbf{a}\times \mathbf{b}& =\left(a_1{b}_2-a_2{b}_1\right)\mathbf{k}& \hfill & \\ \hfill & =\left|\begin{array}{cc}\hfill a_1\hfill & \hfill a_2\hfill \\ \hfill b_1\hfill & \hfill b_2\hfill \end{array}\right|\mathbf{k}.& \hfill & \end{array}$$

Writing the result as a determinant, as we did in the last step, is a handy way to remember the result.

What about the general case where $a_3$ and $b_3$ might not be zero? Well, that’s too many manipulations for me to write it out here, and you probably wouldn’t read it anyway. To calculate the result, it’s just the same process as above. You can verify yourself that in general

$$\begin{array}{llll}\hfill \mathbf{a}\times \mathbf{b}& =\left(a_1\mathbf{i}+a_2\mathbf{j}+a_3\mathbf{k}\right)\times \left(b_1\mathbf{i}+b_2\mathbf{j}+b_3\mathbf{k}\right)& \hfill & \\ \hfill & =\left|\begin{array}{cc}\hfill a_2\hfill & \hfill a_3\hfill \\ \hfill b_2\hfill & \hfill b_3\hfill \end{array}\right|\mathbf{i}-\left|\begin{array}{cc}\hfill a_1\hfill & \hfill a_3\hfill \\ \hfill b_1\hfill & \hfill b_3\hfill \end{array}\right|\mathbf{j}+\left|\begin{array}{cc}\hfill a_1\hfill & \hfill a_2\hfill \\ \hfill b_1\hfill & \hfill b_2\hfill \end{array}\right|\mathbf{k}& \hfill & \end{array}$$

Looking at the formula for the $3\times 3$ determinant, we see that the formula for a cross product looks a lot like the formula for the $3\times 3$ determinant. If we allow a matrix to have the vector $\mathbf{i}$, $\mathbf{j}$, and $\mathbf{k}$ as entries (OK, maybe this doesn’t make sense, but this is just as a tool to remember the cross product), the $3\times 3$ determinant gives a handy mnemonic to remember the cross product:

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

This is the way I remember how to compute the cross product.

Contents by topic

1. Geometry in higher dimensions

1a. Preliminaries

• The cross product
• The component formula for the cross product
• The scalar triple product
• Cross product examples
• Scalar triple product example
• Spherical coordinates
• Examples of n-dimensional vectors
• Function notation

1b. Lines and planes

1c. Matrices

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