We've learned how to add vectors, so it now becomes an issue to learn to multiply them. But the many questions arrise about exactly how to perform such a multiplication.

• What is the rule?
• What kind of output do you get? (Scalar? Vector? Tensor?)
• Is this multiplication Commuttative?

  For the Dot product, the following general properties apply:

  

  Motivation

  Part of the Motivation for using the dot product is the physical situation to which it applies, namely that of computin work done on an object by a given force over a given distance, as shown below:

  

  To help us compute this, we define the dot product as follows:

  

  

  

  Dot products can also be used to help find the equation for a line in 2 dimentional space. Given the line below, with the indicated normal vector N, we can compute it's equation using the knowledge that the dot product of two perpindicular vectors is 0.

  

  N = ai + bj

  P0P=(x-x0)i + (y-y0)j

  N dotted with P0P = 0

  a(x-x0) + b(y-y0)=0

  ax+by = a*x0 + b*y0 = c.

  • Projections

    

    Vector Projections

    

  • Resolve B into B1 (parallel to A) and B2 (perpindicular to A).