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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.

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