Dot Product

As one definition: the Dot Product is the sum of the products of the corresponding entries of the two sequences of numbers.

The example below makes use of this definition:

  • A = {1,2,3}
  • B = {1,3,5}
  • A \cdot B = 1*1+2*3+3*5 = 1 + 6 + 15 = 22

We can express this as:

\displaystyle \sum_{i=1}^n a_i b_i

We learn in Calculus III that we can do a Dot Product on two vectors. The calculation is matrix multiplication on the first vector and the transpose of the second vector:

<1,3,5>^T = \begin{bmatrix} 1 \\3 \\5 \end{bmatrix}

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

There is another calculation for the Dot Product:

a \cdot b = ||a|| \: ||b|| cos \theta

The angle theta is the angle between vector a and vector b.

|| a || is the length of a.

Appendix A

We are interested in the dot product of (a,b) and (-b,a) for something coming…

a(-b) + ba = -ab + ab = 0

Dot Product is related to (possibly synonymous with) Inner Product and Scalar Product.