3D Fractal Still Image
A still frame from a fractal animation I made with Mandelbulber2

Memorize the Cross Product with "Water Flows Under the Bridge"

I've been trying to memorize the formula for computing the components of the cross product, but I had a lot of trouble.

LaTeX for the above image:

\mathbf{A} \times \mathbf{B} = \begin{pmatrix} A_y B_z - A_z B_y \\ A_z B_x - A_x B_z \\ A_x B_y - A_y B_x \end{pmatrix}

All those components look so similar! It was very confusing to remember the order of which components were required for each step.

However I knew that strong visual images can help a lot with memorization, and so I came up with one for the cross product as well. I call it "Water Flows Under the Bridge"

Step 1: Write Out AB-AB for each Component

This one is easy enough to memorize every component has the same format: Component of first vector times component of second vector MINUS Component of first vector times component of second vector. Or, each component goes AB-AB.

Step 2: Build Our X Bridge

Now, we start labeling our x-components, by visualizing a bridge that to help us know where to place the x-components.


And where the bridge hits, we have the x-component:

Step 3: Water Flows Under the Bridge: fill in y and z components downwards from the x

Since water flows under the bridge, we can fill in our y and z components by going downwards from the x-components and fill those in.

The trick here is to remember that if our water flows off screen, and so it just goes right back upwards. And for every letter you hit, you deposit the components in order. Since the alphabet goes x,y,z , you deposit y first and then z, resulting in the equation we started with


(Just to be clear, the first row is the x component, the second row the y component, and the third row is the z component of the final result vector.)

This article was updated on October 8, 2020