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.