Editing Dot product (section)

== Definition ==
The dot product may be defined algebraically or geometrically. The geometric definition is based on the notions of angle and distance (magnitude) of vectors. The equivalence of these two definitions relies on having a [[Cartesian coordinate system]] for Euclidean space.

In modern presentations of [[Euclidean geometry]], the points of space are defined in terms of their [[Cartesian coordinates]], and [[Euclidean space]] itself is commonly identified with the [[real coordinate space]] <math>\mathbf{R}^n</math>. In such a presentation, the notions of length and angle are defined by means of the dot product. The length of a vector is defined as the [[square root]] of the dot product of the vector by itself, and the [[cosine]] of the (non oriented) angle between two vectors of length one is defined as their dot product.  So the equivalence of the two definitions of the dot product is a part of the equivalence of the classical and the modern formulations of Euclidean geometry.

=== Coordinate definition ===
The dot product of two vectors <math>\mathbf{a} = [a_1, a_2, \cdots, a_n]</math> and {{nowrap|1=<math>\mathbf{b} = [b_1, b_2, \cdots, b_n]</math>,}} specified with respect to an [[orthonormal basis]], is defined as:<ref name="Lipschutz2009">{{cite book |author1=S. Lipschutz |author2=M. Lipson |title= Linear Algebra (Schaum's Outlines) | edition= 4th | year= 2009|publisher= McGraw Hill|isbn=978-0-07-154352-1}}</ref>
<math display="block">\mathbf a \cdot \mathbf b = \sum_{i=1}^n a_i b_i = a_1 b_1 + a_2 b_2 + \cdots + a_n b_n</math>
where <math>\Sigma</math> denotes [[summation]] and <math>n</math> is the [[dimension]] of the [[vector space]]. For instance, in [[Three-dimensional space (mathematics)|three-dimensional space]], the dot product of vectors {{nowrap|<math> [1,3,-5] </math>}} and {{nowrap|<math> [4,-2,-1] </math>}} is:
<math display="block">
\begin{align}
\ [1, 3, -5] \cdot [4, -2, -1] &= (1 \times 4) + (3\times-2) + (-5\times-1) \\
&= 4 - 6 + 5 \\
&= 3
\end{align}
</math>

Likewise, the dot product of the vector {{nowrap|<math>[1,3,-5]</math>}} with itself is:
<math display="block">
\begin{align}
\ [1, 3, -5] \cdot [1, 3, -5] &= (1 \times 1) + (3\times 3) + (-5\times -5) \\
&= 1 + 9 + 25 \\
&= 35
\end{align}
</math>

If vectors are identified with [[column matrix|column vectors]], the dot product can also be written as a [[matrix multiplication|matrix product]]
<math display="block">\mathbf a \cdot \mathbf b = \mathbf a^{\mathsf T} \mathbf b,</math>
where <math>\mathbf a{^\mathsf T}</math> denotes the [[transpose]] of <math>\mathbf a</math>.

Expressing the above example in this way, a 1 × 3 matrix ([[row vector]]) is multiplied by a 3 × 1 matrix ([[column vector]]) to get a 1 × 1 matrix that is identified with its unique entry:
<math display="block">
\begin{bmatrix}
  1 & 3 & -5
\end{bmatrix}
\begin{bmatrix}
  4 \\ -2 \\ -1
\end{bmatrix} = 3 \, .
</math>

=== 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.

=== 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.

=== Equivalence of the definitions ===
If <math>\mathbf{e}_1,\cdots,\mathbf{e}_n</math> are the [[standard basis|standard basis vectors]] in <math>\mathbf{R}^n</math>, then we may write
<math display="block">\begin{align}
\mathbf a &= [a_1 , \dots , a_n] = \sum_i a_i \mathbf e_i \\
\mathbf b &= [b_1 , \dots , b_n] = \sum_i b_i \mathbf e_i.
\end{align}
</math>
The vectors <math>\mathbf{e}_i</math> are an [[orthonormal basis]], which means that they have unit length and are at right angles to each other.  Since these vectors have unit length,
<math display="block"> \mathbf e_i \cdot \mathbf e_i = 1 </math>
and since they form right angles with each other, if <math>i\neq j</math>,
<math display="block"> \mathbf e_i \cdot \mathbf e_j = 0 .</math>
Thus in general, we can say that:
<math display="block"> \mathbf e_i \cdot \mathbf e_j = \delta_ {ij} ,</math>
where <math>\delta_{ij}</math> is the [[Kronecker delta]].

[[File:Skalarprodukt geometrisch.svg|thumb|upright=1.0|Vector components in an orthonormal basis]]

Also, by the geometric definition, for any vector <math>\mathbf{e}_i</math> and a vector <math>\mathbf{a}</math>, we note that
<math display="block"> \mathbf a \cdot \mathbf e_i = \left\| \mathbf a \right\| \left\| \mathbf e_i \right\| \cos \theta_i = \left\| \mathbf a \right\| \cos \theta_i = a_i ,</math>
where <math>a_i</math> is the component of vector <math>\mathbf{a}</math> in the direction of <math>\mathbf{e}_i</math>. The last step in the equality can be seen from the figure.

Now applying the distributivity of the geometric version of the dot product gives
<math display="block"> \mathbf a \cdot \mathbf b = \mathbf a \cdot \sum_i b_i \mathbf e_i = \sum_i b_i ( \mathbf a \cdot \mathbf e_i ) = \sum_i b_i a_i= \sum_i a_i b_i ,</math>
which is precisely the algebraic definition of the dot product.  So the geometric dot product equals the algebraic dot product.