The second-most important use of a corkscrew

Graphic - corkscrew

The direction of a cross product is defined in two stages.

First it is a direction that is perpendicular to the plane through the two vectors: so the cross product of two vectors is always perpendicular to both. This isn’t enough of a definition: there are always two directions that can claim to be perpendicular to a plane (eg ‘into’ and ‘out of’ this page). Here is where we need the ‘corkscrew’ of the title.

Take the corkscrew, in your mind’s eye, and put its tip on the point at which the two vectors meet (We’ll assume they are drawn with a common origin.) Align it perpendicular to their plane (either into or out of the plane, it doesn’t matter which). Now turn the corkscrew in such a way as to ‘rotate $\vec{a}$ into $\vec{b}$’, taking the shortest route. The direction in which the corkscrew advances gives the direction of the vector cross product $\vec{a} \times \vec{b}$.

This sounds harder than it is: bring a corkscrew along to a tutorial for a practical demonstration if you are confused.

The following applet allows you to explore how the vector $\vec{C}$ defined by

\[ \vec{C} =\vec{A} \times \vec{B} \]
varies in both magnitude and direction as you vary $\vec{A}$ or $\vec{B}$.

Dragging with the mouse in the applet area rotates the axes.

The vector $\vec{C}$ is not drawn to the same scale as $\vec{A}$ and $\vec{B}$.