# Dot (Inner or Scalar) Product

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).

Notes by Lawrence C. Weintraub 1-29-95