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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.

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