Day 59 in MAT229

• We're going to hold it at the time slot for the TR classes starting at 1:40, which is Thursday, 5/6, 12:20-2:20. It will be held on Canvas, but you will have two hours to do it.

• We are open to allowing those who would prefer taking it in the TR at 9:25 slot (which is Tuesday, 5/4, 10:10-12:10) to do so. Let me know if this is you.

• You will be allowed a page of notes and a calculator.

The cross product ${\overline{u}}\times{\overline{v}}$ is a vector product (produces a vector) -- but only works in three-dimensions. Its magnitude reflects the extent to which the two vectors ${\overline{u}}$ and ${\overline{v}}$ are parallel. In particular, the cross product has 0 length when two vectors point in parallel directions. Thus it provides a test for parallelness.

It also is used to compute the area of a parallelogram formed by putting the butts of the two vectors together: \[ |{\overline{u}}\times{\overline{v}}|=|{\overline{u}}||{\overline{v}}| \sin(\theta) \] Notice that the parallelogram has zero area when the vectors are parallel, and maximal area when the vectors are perpendicular.

Finally, the cross product provides a means for producing a vector perpendicular to both ${\overline{u}}$ and ${\overline{v}}$ -- so enables us to "leap from a plane", as it were, the plane in which that parallelogram, formed by putting the butts of the two vectors together, lives.

Since I borrowed this image I shouldn't be too critical, but it's important to note that the unit vector mentioned, $\hat{a}_n$, is the unit vector chosen by the right-hand rule (more about that in the materials below). I believe that the use of the subscript "$n$" on that unit vector indicates "normal" -- that is, perpendicular -- to the plane.

• Cross Products (.nb)
• Cross Products (.pdf)
• Long talks you through cross products.

• From Stewart's text
• Supporting Mathematica code