Math Insight

The cross product

 

We define the cross product of two three-dimensional vectors $\vc{a}$ and $\vc{b}$ by the requirements:

  • $\vc{a} \times \vc{b}$ is a vector that is perpendicular to both $\vc{a}$ and $\vc{b}$.
  • $\|\vc{a} \times \vc{b}\|$ is the area of the parallelogram spanned by $\vc{a}$ and $\vc{b}$ (i.e. the parallelogram whose adjacent sides are the vectors $\vc{a}$ and $\vc{b}$).
  • The direction of $\vc{a} \times \vc{b}$ is determined by the right-hand rule. (This means that if we curl the fingers of the right hand from $\vc{a}$ to $\vc{b}$, then the thumb points in the direction of $\vc{a} \times \vc{b}$.)

From simple trigonometry, we can calculate that the area of the parallelogram spanned by $\vc{a}$ and $\vc{b}$ is \begin{align*} \|\vc{a}\| ~ \|\vc{b}\| \sin\theta, \end{align*} where $\theta$ is the angle between $\vc{a}$ and $\vc{b}$. (One can view the parallelogram as having a base of length $\|\vc{b}\|$ and perpendicular height $\|\vc{a}\| \sin \theta$, as pictured below.)

Area of parallelogram

This formula shows that the magnitude of the cross product is largest when $\vc{a}$ and $\vc{b}$ are perpendicular. On the other hand, if $\vc{a}$ and $\vc{b}$ are parallel or if either vector is the zero vector, then the cross product is the zero vector. (It is a good thing that we get the zero vector in these cases so that the above definition still makes sense. If the vectors are parallel or one vector is the zero vector, then there is not a unique line perpendicular to both $\vc{a}$ and $\vc{b}$. But since there is only one vector of zero length, the definition still uniquely determines the cross product.)

Below is an applet that helps illustrate how the cross product works. Although it is admittedly hard to manipulate in a precise manner, you can convince yourself that the above properties of the cross product are satisfied by the cross product vector shown in the applet.

Applet: Cross product

The Java applet did not load, and the above is only a static image representing one view of the applet. The applet was created with LiveGraphics3D. The applet is not loading because it looks like you do not have Java installed. You can click here to get Java.

Applet: Cross product

Cross product. The vector $\color{red}{\vc{c}}$ (in red) is the cross product of $\color{blue}{\vc{a}}$ (in blue) and $\color{green}{\vc{b}}$ (in green), $\color{red}{\vc{c}} = \color{blue}{\vc{a}} \times \color{green}{\vc{b}}$. The parallelogram formed by $\vc{a}$ and $\vc{b}$ is shown in cyan. Using the mouse, you can drag the balls at the ends of the vectors $\color{red}{\vc{c}}$ and $\color{green}{\vc{b}}$ to change these vectors. See how the cross product $\color{red}{\vc{c}}$ and the parallelogram change in response. The three-dimensional perspective of this graph is hard to perceive when the graph is still, but you may see it better if you keep the figure rotating.

More information about applet.

Notice that the area of the parallelogram (and hence the magnitude of the cross product) go to zero as $\vc{a}$ and $\vc{b}$ approach parallel (where the term “parallel” also includes what you might think as anti-parallel). You can also verify that the applet demonstrates $\vc{b} \times \vc{a} = - \vc{a} \times \vc{b}$ and $\vc{a} \times \vc{a} = \vc{0}$, which are important properties of the cross product.

The geometric definition of the cross product is nice for understanding its properties. However, it's not too convenient for numerically calculating the cross product of vectors given in terms of their coordinates. For such calculations, you may want to have the formula for the cross product in terms of components. With the cross product formula in hand, you can make short work of any calculation, as illustrated by these examples of calculating cross products and areas of parallelograms.