Definition of Dot Products

The dot product is an operation on 2 \(n\)-dimensional vectors. The formula is \[< a_1, a_2, \dots, a_n > \cdot < b_1, b_2, \dots, b_n > = a_1b_1 + a_2b_2 + \dots + a_nb_n\]

The dots product is computed by multiplying cooridinate-wise, then adding all the products.

For example, here is the dot product of 3-dimensional vectors: \begin{align} <2,1,4> \cdot <3,5,2> & = 2 \cdot 3 + 1 \cdot 5 + 4 \cdot 2 \\ & = 6 + 5 + 8 \\ & = 19 \end{align} Here is an example with 2-dimensional vectors: \[<2,3> \cdot <-1, 0> = -2 + 0 = -2\] The dot product is not defined for vectors of different dimensions.