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## Lecture (20 minutes)

**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:
- Volunteer 1: Hold up your (blank) small white board in the air at some angle.
- Volunteer 2: Without turning around, hold up your small white board in the same orientation as Volunteer 1.
- One way would be to each draw two vectors on your whiteboard and align them.
- A better way is to specify the normal vector.
- How do you specify which way to hold the white surface?
- 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.