Editing Dot product (section)

=== Scalar projection and first properties ===
[[File:Dot Product.svg|thumb|right|Scalar projection]]
The [[scalar projection]] (or scalar component) of a Euclidean vector <math>\mathbf{a}</math> in the direction of a Euclidean vector <math>\mathbf{b}</math> is given by
<math display="block"> a_b = \left\| \mathbf a \right\| \cos \theta ,</math>
where <math>\theta</math> is the angle between <math>\mathbf{a}</math> and <math>\mathbf{b}</math>.

In terms of the geometric definition of the dot product, this can be rewritten as
<math display="block">a_b = \mathbf a \cdot \widehat{\mathbf b} ,</math>
where <math> \widehat{\mathbf b} = \mathbf b / \left\| \mathbf b \right\| </math> is the [[unit vector]] in the direction of <math>\mathbf{b}</math>.

[[File:Dot product distributive law.svg|thumb|right|Distributive law for the dot product]]
The dot product is thus characterized geometrically by<ref>{{cite book | last1=Arfken | first1=G. B. | last2=Weber | first2=H. J. | title=Mathematical Methods for Physicists | publisher=[[Academic Press]] | location=Boston, MA | edition=5th | isbn=978-0-12-059825-0 | year=2000 | pages=14–15 }}</ref>
<math display="block"> \mathbf a \cdot \mathbf b = a_b \left\| \mathbf{b} \right\| = b_a \left\| \mathbf{a} \right\| .</math>
The dot product, defined in this manner, is [[Homogeneous function|homogeneous]] under scaling in each variable, meaning that for any scalar <math>\alpha</math>,
<math display="block"> ( \alpha \mathbf{a} ) \cdot \mathbf b = \alpha ( \mathbf a \cdot \mathbf b ) = \mathbf a \cdot ( \alpha \mathbf b ) .</math>
It also satisfies the [[distributive law]], meaning that
<math display="block"> \mathbf a \cdot ( \mathbf b + \mathbf c ) = \mathbf a \cdot \mathbf b + \mathbf a \cdot \mathbf c .</math>

These properties may be summarized by saying that the dot product is a [[bilinear form]]. Moreover, this bilinear form is [[positive definite bilinear form|positive definite]], which means that <math> \mathbf a \cdot \mathbf a </math> is never negative, and is zero if and only if <math> \mathbf a = \mathbf 0 </math>, the zero vector.