## Properties of the Dot or Scalar Product

The dot product has certain useful properties.
The dot product is a number.
$\vec{a} \cdot \vec{b}= \vec{b} \cdot \vec{a}$
.
This is consistent with the formula for the angle between
$\vec{a}, \: \vec{b}$

$cos \theta = \frac{\vec{a} \cdot \vec{b}}{\| vec{a} \| \| \vec{b} \| }$
, sin
$cos (\theta)=cos(- \theta)$
so the cos of the angle from
$\vec{a}$
to
$\vec{b}$
is the same as the cos of the angle from
$\vec{b}$
to
$\vec{a}$
.
The dot product is distributive.
$\vec{a} \cdot (\vec{b}+ \vec{c})=\vec{a} \cdot \vec{b}+\vec{a} \cdot \vec{c}$
.
If
$\theta$
is acute then
$\vec{a} \cdot \vec{b} \gt 0$
and if
$\theta$
is obtuse then
$\vec{a} \cdot \vec{b} \lt 0$
.
If
$\vec{a} \cdot \vec{b}= 0$
then
$cos \theta =0$
and
$\vec{a}, \: \vec{b}$
are perpendicular.
If
$\vec{a} \cdot \vec{b}= \| \vec{a} \| \| \vec{b} \|$
then
$cos \theta =1$
and
$\vec{a}, \: \vec{b}$
are parallel.
If
$\vec{a} \cdot \vec{b}=- \| \vec{a} \| \| \vec{b} \|$
then
$cos \theta =180^o$
and
$\vec{a}, \: \vec{b}$
are anti parallel.