Dot Product

A dot product is a math operation on two lists with equal lengths. Equivalent values form products and these products become the terms in a summation.

(1,2,3) \cdot (4,5,6)

1*4 + 2*5 + 3*6

4 + 10 + 18

32

If we are working with vectors, we need the first vector to be a row vector and the second vector to be a column vector (see Appendix E).

\begin{bmatrix}1 & 2 & 3 \end{bmatrix} \begin{bmatrix}4 \\ 5 \\ 6 \end{bmatrix}

Appendix A

An Inner Product is a dot product if we are in a finite dimensional Euclidean Space.

The dot product is NOT the only Inner Product that can be defined on Euclidean Space.

Appendix B

The dot product might be one of the first places where we should show Einstein notation.

A=(a_1,a_2,a_3)

B=(b_1,b_2,b_3)

\displaystyle A \cdot B = \sum_i a_i \: b_i

A \cdot B = a_i \: b_i

Because i occurs twice in a term we know there is a summation over i in that term.

Appendix C

The example below shows how every calculation in a matrix multiplication is a dot product.

\begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix} \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \end{bmatrix} =

\begin{bmatrix} \begin{bmatrix} 1 & 2 \end{bmatrix} \cdot \begin{bmatrix} 1 \\ 4 \end{bmatrix} & \begin{bmatrix} 1 & 2 \end{bmatrix} \cdot \begin{bmatrix} 2 \\ 5 \end{bmatrix} & \begin{bmatrix} 1 & 2 \end{bmatrix} \cdot \begin{bmatrix} 3 \\ 6 \end{bmatrix} \\ \\ \begin{bmatrix} 3 & 4 \end{bmatrix} \cdot \begin{bmatrix} 1 \\ 4 \end{bmatrix} & \begin{bmatrix} 3 & 4 \end{bmatrix} \cdot \begin{bmatrix} 2 \\ 5  \end{bmatrix} & \begin{bmatrix} 3 & 4 \end{bmatrix} \cdot \begin{bmatrix} 3 \\ 6 \end{bmatrix} \end{bmatrix}

Appendix D

a \cdot b = a_1b_1 + a_2b_2 + a_3b_3

The above is only true if we are working with orthonormal basis vectors.

Appendix E

In a paper about tensors, the authors informed us that they decided to reserve the dot product for use only for work with metric tensors and inverse metric tensors.

We don’t know if this was specific to this paper, or if this is a tradition that all mathematicians everywhere observe.

Appendix F

Someone is going to notice – – this rule only makes sense if you say we’re doing matrix multiplication.

It might be good to admit it, because what is happening is, we’re starting with vectors and then moving on to matrices and developing rules and then we’re going to say “Push that big G button and generalize all this” so that we can use it in higher stuff.