Changes

: <math>\vec{x} \cdot \vec{y} = \sum_{i=1}^n x_i y_i = x_1 y_1 + \cdots +x_n y_n </math>

: <math>\vec{x} \cdot \vec{y} = \sum_{i=1}^n x_i y_i = x_1 y_1 + \cdots +x_n y_n </math>

Unlike the [[cross product]], the dot product is a [[scalar]], not a vector, and has no direction.

Unlike the [[cross product]], the dot product is a [[scalar]], not a vector, and has no direction.  Also, unlike the cross product, the dot product is [[commutative]].

For example, in 3-dimensional Euclidean space, let <math>\vec{x} = \langle 1, 2, 3 \rangle</math> and <math>\vec{y} = \langle 4, 5, 6\rangle</math>.

For example, in 3-dimensional Euclidean space, let <math>\vec{x} = \langle 1, 2, 3 \rangle</math> and <math>\vec{y} = \langle 4, 5, 6\rangle</math>.