Note: Different students likely have different ways of using the determinate of a matrix to perform a cross product; consider allowing them to use whatever method they are comfortable with but make sure student's who are not comfortable with determinates are not left behind.

• The cross product
  • What kind of objects are you multiplying?
  • What kind of object do you get?
  • What happens if you go in reverse order?
  • What right hand rule do you want to use?
• Two brave volunteers to the front of the room, back to back on opposite sides:
  1. Volunteer 1: Hold up your (blank) small white board in the air at some angle.
  2. Volunteer 2: Without turning around, hold up your small white board in the same orientation as Volunteer 1.
  3. One way would be to each draw two vectors on your whiteboard and align them.
  4. A better way is to specify the normal vector.
  5. How do you specify which way to hold the white surface?
  6. How do you specify how big your surface is (its area)?
• The triple product
  • SWBQ: Now suppose you want to find the volume of your whiteboard (it has nonzero thickness).
    • Tell students to work with their neighbor after a minute or so.
  • If you take the dot product between a vector and the cross product of two other vectors, you find the volume of the parellelpiped defined by the three vectors.
    • Interestingly, it doesn't matter which vectors you pick to be in the cross product, but the order of the vectors does matter, and the order is cyclical.
• Now you are going to do this for different surfaces in curvilinear coordinates.