The Vector Cross Product
- A JAVA Interactive Tutorial

To use this tutorial, you will need a "JAVA-enabled" web browser.

Just below you will see a drawing; the vector labeled c is being calculated according to your specifications for the vectors a and b. You move these by clicking on their tips and dragging them around the plane with the mouse. You can also "spin the plane" by clicking and dragging on other parts of the picture.

To See and To Do

• Line up b so that it is parallel to a. What happens to c?
• Line up b so that it's perpendicular to a. Now what happens to c?
• Keeping b perpendicular to a, shorten b. What happens to c?
• Play around with a or b - can you make c drop underneath the plane?
• Spin the plane; does c seem to be perpendicular to it?

What's Going On?

In the drawing you see three vectors a, b, and c. The vector c is calculated from the vectors a and b using the vector cross product. In mathematical notation you write:
c = a x b

Magnetic forces are described using the vector cross product. So are the "torques" governing spinning and rotating objects. And there are still more examples.

It is important to understand that any two vectors a and b lie in some plane. This is the plane indicated in the applet's output. The vector c calculated using the cross-product rules is always perpendicular (or "normal") to this plane. To help you understand these relationships, you can spin the plane shown in the figure. Place the mouse on some spot on the plane, click, and move the mouse around. The plane and the three vectors all rotate to accomodate the new perspective. We'll discuss how to figure out if c points "up" or "down" later on.

Mathematics of the Vector Cross Product

Vectors have been denoted here using boldfaced characters. We denote the magnitude (or length) of the vector a as just "a" (without boldface). Note that a magnitude is always a positive number.
The magnitude c of the cross product is given by the formula c = ab sin(\phi), where \phi is the angle included between a and b.

The direction of c has only been defined as perpendicular to the plane of a and b. To determine whether c points up or down from the plane, you use a "right-hand" rule.

The right hand rule is difficult to explain in words, but try the following exercise. Point the four fingers of your right hand so that they point in the same direction as the vector a. Now get your hand organized so that you can swing the palm of your hand from a to b. If you succeeded, then your thumb will point along c. See if you can make sense of this right-hand rule by checking it against the applet's output.