Product Operations on Vectors

The first three sections were about the linear operations between vectors — addition, subtraction and scalar multiplication — and they only change the length and the direction of a vector. This section turns to a different class of operation: “multiplying” two vectors to get a new quantity. The scalar product gives a scalar (the product of the lengths and the angle), the vector product gives a new vector perpendicular to both, and the mixed product combines the two into a tool for testing coplanarity. None of them are linear operations, so they cannot be derived from the axioms of the earlier sections; they have to be set up separately.