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.

Properties of the Dot Product

• Commutativity: Given vectors \(\overrightarrow{u}\) and \(\overrightarrow{v}\) of the same dimension, \[\overrightarrow{u} \cdot \overrightarrow{v} = \overrightarrow{v} \cdots \overrightarrow{u}\]
• Associativity: Given vectors \(\overrightarrow{u},\) \(\overrightarrow{v},\) and \(\overrightarrow{w}\) of the same dimension, \[(\overrightarrow{u} \cdot \overrightarrow{v}) \cdot \overrightarrow{w} = \overrightarrow{u} \cdot (\overrightarrow{v} \cdot \overrightarrow{w})\]
• Distributivity: Given vectors \(\overrightarrow{u},\) \(\overrightarrow{v},\) and \(\overrightarrow{w}\) of the same dimension, \[\overrightarrow{u} \cdot (\overrightarrow{v} + \overrightarrow{w}) = \overrightarrow{u} \cdot \overrightarrow{v} + \overrightarrow{u} \cdot \overrightarrow{w}\]
• Zero element: Given a vector \(\overrightarrow{v}\) and the zero vector \(\overrightarrow{0}\) of the same dimension, \[x(\overrightarrow{v}+\overrightarrow{w}) = x\overrightarrow{v}+x\overrightarrow{w}.\]
• Associativity with scalars: Given a scalar \(x\) and vectors \(\overrightarrow{u}\) and \(\overrightarrow{v}\) of the same dimension, \[x(\overrightarrow{u} \cdot \overrightarrow{v})=(x\overrightarrow{u}) \cdot \overrightarrow{v}.\]