## Properties of the Vector or Cross Product

The vector or cross product of two vectors has several useful properties.
The vector product of two vectors is itself a vector, and is perpendicular to the two vectors.
If the ngle between vectors
$\vec{a}$
and
$\vec{b}$
is
${}+90^o$
so that vector
$\vec{b}$
is 90 degrees anticlockwise from
$\vec{a}$
, then the vectors
$\vec{a}, \: \vec{b}, \: \vec{a} \times \vec{b}=\vec{c}$
forms a right handed coordinates system.
$\vec{a} \times k \vec{a}=0$

$\vec{a} \times \vec{b}=\| \vec{a} \| \| \vec{b}, \| \vec{a}. \| vec{b} \neq \vec{0} \|$
if
$\vec{a}, \: \vec{b}$
are perpendicular, and vice versa.
$\vec{a} \times \vec{b}= - \vec{b} \times \vec{a}$

$\| \vec{a} \times \vec{b} \|= \| \vec{b} \times \vec{a} \|$

$\vec{a} \cdot (\vec{b} \times \vec{b})=\left| \begin{array}{ccc} a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \\ c_1 & c_2 & c_3 \end{array} \right|$
is the vector triple product. It is a number, with magnitude equal to the volume of the parallelpiped formed by the vectors
$\vec{a}, \: \vec{b}, \: \vec{c}$
.
$\vec{a} \times (\vec{b}+\vec{c})=(\vec{a} \times \vec{b})+(\vec{a} \times \vec{c})$

$(\vec{a}+\vec{b}) \times (\vec{c}+\vec{d})=(\vec{a} \times \vec{c})+(\vec{a} \times \vec{d})+(\vec{b} \times \vec{c})+(\vec{b} \times \vec{d})$