Answer to Question #206678 in Linear Algebra for Snakho

Question

Prove that the dot between two vectors is commutative not associative

Answer 1

The dot product of two Euclidean vectors $\vec a$  and $\vec b$  is defined by

$\vec a \cdot \vec b=||\vec a||\ ||\vec b|| \cos \theta$

where $\theta$  is the angle between $\vec a$ and $\vec b.$

The dot between two vectors is commutative. This follows from the definition of the dot product

$\vec a \cdot \vec b=||\vec a||\ ||\vec b|| \cos \theta\ =||\vec b||\ ||\vec a|| \cos \theta=\vec b \cdot \vec a$

where $\theta$ is the angle between $\vec a$ and $\vec b.$

The dot product between a scalar $(\vec a \cdot \vec b)$ ) and a vector $(\vec c)$ is not defined, which means that the expressions involved in the associative property, $(\vec a \cdot \vec b)\cdot \vec c$  or $\vec a \cdot (\vec b\cdot \vec c)$ are both ill-defined.

Hence the dot between two vectors is not associative.