Editing Dot product (section)

=== Geometric definition ===
[[File:Inner-product-angle.svg|thumb|Illustration showing how to find the angle between vectors using the dot product]]
[[File:Tetrahedral angle calculation.svg|thumb|216px|<!-- specify width as minus sign vanishes at most sizes --> Calculating bond angles of a symmetrical [[tetrahedral molecular geometry]] using a dot product]]
In [[Euclidean space]], a [[Euclidean vector]] is a geometric object that possesses both a magnitude and a direction.  A vector can be pictured as an arrow.  Its magnitude is its length, and its direction is the direction to which the arrow points. The [[Magnitude (mathematics)|magnitude]] of a vector <math>\mathbf{a}</math> is denoted by <math> \left\| \mathbf{a} \right\| </math>. The dot product of two Euclidean vectors <math>\mathbf{a}</math> and <math>\mathbf{b}</math> is defined by<ref name="Spiegel2009">{{cite book |author1=M.R. Spiegel |author2=S. Lipschutz |author3=D. Spellman |title= Vector Analysis (Schaum's Outlines)|edition= 2nd |year= 2009|publisher= McGraw Hill|isbn=978-0-07-161545-7}}</ref><ref>{{cite book|author1=A I Borisenko|author2=I E Taparov|title=Vector and tensor analysis with applications | publisher=Dover | translator=Richard Silverman | year=1968 | page=14}}</ref><ref name=":1" />
<math display="block">\mathbf{a}\cdot\mathbf{b}= \left\|\mathbf{a}\right\| \left\|\mathbf{b}\right\|\cos\theta ,</math>
where <math>\theta</math> is the [[angle]] between <math>\mathbf{a}</math> and <math>\mathbf{b}</math>.

In particular, if the vectors <math>\mathbf{a}</math> and <math>\mathbf{b}</math> are [[orthogonal]] (i.e., their angle is <math>\frac{\pi}{2}</math> or <math>90^\circ</math>), then <math>\cos \frac \pi 2 = 0</math>, which implies that
<math display="block">\mathbf a \cdot \mathbf b = 0 .</math>
At the other extreme, if they are [[codirectional]], then the angle between them is zero with <math>\cos 0 = 1</math> and
<math display="block">\mathbf a \cdot \mathbf b = \left\| \mathbf a \right\| \, \left\| \mathbf b \right\| </math>
This implies that the dot product of a vector <math>\mathbf{a}</math> with itself is
<math display="block">\mathbf a \cdot \mathbf a = \left\| \mathbf a \right\| ^2 ,</math>
which gives
<math display="block"> \left\| \mathbf a \right\| = \sqrt{\mathbf a \cdot \mathbf a} ,</math>
the formula for the [[Euclidean length]] of the vector.