Cross product discovery applet

The cross product of two vectors u and v in R³ is a vector in R³ that is perpendicular to both u and v. It is denoted uxv. Use the applet below to discover the basic properties of the cross product. The questions below the applet are intended to guide your discovery.

Note: in the applet below, the vectors u and v live in R³. The green plane contains the two vectors u and v and uxv is perpendicular to the green plane. "Up" from the green plane corresponds to "out" of the computer display.

Discovery questions:

• Is uxv a vector or a scalar?
• Is the cross product commutative? That is, is uxv=vxu?
• When does uxv point out of the screen and when does it point into the screen?
• When is uxv the zero vector?
• What happens to uxv when u or v gets longer?
• What is (uxv)•u? (uxv)•v?
• What is |uxv| when u and v are orthogonal vectors?

Check the box labeled "Show the unit circle."

• Fix one vector on the unit circle. Move the other vector around the circle and observe |uxv|. |uxv| can be expressed as a familiar, elementary function. What do you think it is?
• Is this still true if u and v aren't unit vectors?

Check the box labeled "Show parallelogram."

• What do you notice?

Created by David Richeson using GeoGebra.