Inner Product

In a real vector space, something that is an inner product < , > will satisfy the following criteria. For the vectors u,v and w and the scalar, a:

<u + v, w) = <u, w> + <v, w>

<av, w> = a<v, w>

<v, w> = <w, v>

<v, v> > 0 except for when v=0; if v=0 then <v, v> = 0

The dot product relates to cosine through the following equation:

cos \: \theta = \dfrac {x \cdot y}{||x|| \: ||y||}

Appendix A

An Inner Product Space is a Vector Space with Lengths and Angles.

Appendix B

The inner product of two vectors is a scalar.

The inner product of a vector for Force and a vector for Velocity gives us a scalar for Power.

As a check of the units, force is Newtons and Velocity is m/s:

\dfrac {Nm} {s} = \dfrac {J} {s}

The definition of Power is Joules per Second.

Appendix C

If we have two vectors and we express them using the same basis vectors, we can calculate the inner product using the components.

  • a = (a_1, a_2, a_3, ..., a_n)
  • b = (b_1, b_2, b_3, ..., a_n)

Appendix D

The Inner Product is the generalization of the Dot Product.