This worksheet illustrates the geometric significance of the cross product. Move the yellow points to adjust the vectors.
What does the vector along the [math]z[/math]-axis represent?
The cross product [math]\vec{u}\times\vec{v}[/math].
Note that [math]\vec{u}[/math] and [math]\vec{v}[/math] are confined to the [math]xy[/math]-plane in this worksheet. How can we be sure that [math]\vec{u}\times\vec{v}[/math] will always point along the [math]z[/math]-axis?
Because the cross product of two vectors is always perpendicular to each of the vectors.
What does the equality of the two numerical values signify?
The magnitude of the cross product of two vectors is equal to the area of the parallelogram that the vectors span.
Adjust [math]\vec{u}[/math] and [math]\vec{v}[/math] until their cross product points downward. Now, how big is the [i]smallest positive[/i] angle from [math]\vec{u}[/math] to [math]\vec{v}[/math]?