\(\def\pd{\partial}\def\Real{\mathbb{R}}

\)

- There are two main types of products between vectors in \(\mathbb{R}^3\):

The inner/scalar/dot product

$$ A\cdot B = A_x+B_x + A_yB_y + A_z Bz \in \mathbb{R} $$

is commutative, distributive, and homogenenous.

The vector (cross) product:

$$ A\times B = \begin{pmatrix}

A_yB_z-A_zB_y \\

A_zB_c-A_zB_z \\

A_xB_y-A_yB_x\end{pmatrix} \in \mathbb{R^3} $$

is homogeneous, not commutative, not associative, but linear with

respect to each entry. - The cross product \(A\times B\) is always perpendicular to \(A\) and \(B\),

and its length equals to \(|A||B|\sin(\phi)\) where \(\phi\) is the angle

between the vectors.