Difference between revisions of "Dot product"
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{{Navigation|before=[[Simple arithmetic operations]]|overview=[[Vector algebra]]|next=[[Cross product]]}} | {{Navigation|before=[[Simple arithmetic operations]]|overview=[[Vector algebra]]|next=[[Cross product]]}} | ||
+ | <table style="width:100%"><td style="width:50%">{{Exercise|Selftest: Dot product}}</td><td style="width:50%">{{Matlab|MATLAB: Dot product}}</td></table> | ||
[[File:vectoralgebra_dotproduct.jpg|right|500px]] | [[File:vectoralgebra_dotproduct.jpg|right|500px]] | ||
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\vec{\mathbf{a}} \cdot \vec{\mathbf{b}} = a \vec{\mathbf{e}}_{a} \cdot b\vec{\mathbf{e}}_{b} = a b \cos \alpha | \vec{\mathbf{a}} \cdot \vec{\mathbf{b}} = a \vec{\mathbf{e}}_{a} \cdot b\vec{\mathbf{e}}_{b} = a b \cos \alpha | ||
</math> | </math> | ||
− | where <math>\alpha</math> describes the angle between the two vectors which ranges from <math>0</math> to <math>\pi</math> (see figure). The dot product is denoted with a simple point between the vectors or without any sign | + | where <math>\alpha</math> describes the angle between the two vectors which ranges from <math>0</math> to <math>\pi</math> (see figure). The dot product is denoted with a simple point between the vectors or without any sign. |
− | |||
− | + | Regarding the right side of the above equation, the following correlation can be noted: If you project the vector <math>\vec{\mathbf{b}}</math> on the vector <math>\vec{\mathbf{a}}</math>, you get the distance <math>b\cos\alpha</math>. As a consequence the result of the dot product can be seen as the area of a rectangle with the side legths <math>a</math> and <math>b\cos\alpha</math>. The projection can also be done contrariwise (projection of vector <math>\vec{\mathbf{a}}</math> on vector <math>\vec{\mathbf{b}}</math>). So that you get the distance <math>a\cos\alpha</math>. The multiplication of this term with <math>b</math> leads to a rectangle with equivalent area but different aspect ratio (see figure). | |
+ | |||
+ | Another possibility to compute the dot product is to multiply the corresponding components and sum them up: | ||
:<math> | :<math> | ||
− | \vec{\ | + | \vec{\mathbf{a}} \cdot \vec{\mathbf{b}} = a b\cos\alpha = a_x b_x + a_y b_y + a_z b_z |
</math> | </math> | ||
− | + | In general the dot product of n-dimensional vectors is computed as follows: | |
:<math> | :<math> | ||
− | \vec{\ | + | \vec{\mathbf{a}} \cdot \vec{\mathbf{b}} = a b \cos\alpha = \sum_{i=1}^{n} a_i b_i |
</math> | </math> | ||
− | + | which is nothing else than the [[Multiplication of matrices|matrix product]] of the transpose of the first vector with the second vector denoted in [[Matrices|matrix algebra]]:<br/><br/> | |
:<math> | :<math> | ||
− | \vec{\ | + | \vec{\mathbf{a}} \cdot \vec{\mathbf{b}} = \vec{\mathbf{a}}^T \vec{\mathbf{b}} = |
+ | \left[\begin{array}{ccc} | ||
+ | a_1 & \dots & a_n | ||
+ | \end{array}\right] | ||
+ | \left[\begin{array}{c} | ||
+ | b_1 \\ | ||
+ | \vdots \\ | ||
+ | b_n | ||
+ | \end{array}\right] = | ||
+ | \sum_{i=1}^{n} a_i b_i | ||
</math> | </math> | ||
− | + | On the basis of the described relations it appears, that the commutative law holds: | |
+ | :<math> | ||
+ | \vec{\mathbf{a}} \cdot \vec{\mathbf{b}} = \vec{\mathbf{b}} \cdot \vec{\mathbf{a}} | ||
+ | </math> | ||
+ | Furthermore the following special cases can be considered, that often lead to simplifications in technical context: | ||
:<math> | :<math> | ||
\begin{align} | \begin{align} | ||
− | \vec{\ | + | \vec{\mathbf{a}} \cdot \vec{\mathbf{b}} &= ab& |
− | &\text{ | + | &\text{if}& |
− | \vec{\ | + | \vec{\mathbf{a}} & \upuparrows \vec{\mathbf{b}}& &(\text{because} \cos(0) = 1)\\ |
− | \vec{\ | + | \vec{\mathbf{a}} \cdot \vec{\mathbf{b}} &= 0& |
− | &\text{ | + | &\text{if}& |
− | \vec{\ | + | \vec{\mathbf{a}} &\perp \vec{\mathbf{b}}& &(\text{because} \cos(\frac{\pi}{2}) = 0)\\ |
− | \vec{\ | + | \vec{\mathbf{a}} \cdot \vec{\mathbf{b}} &= -ab& |
− | &\text{ | + | &\text{if}& |
− | \vec{\ | + | \vec{\mathbf{a}} &\downarrow\uparrow \vec{\mathbf{b}}& &(\text{because} \cos(\pi) = -1) |
\end{align} | \end{align} | ||
− | </math> | + | </math><br/><br/> |
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− | </ | ||
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{{Multimedia|Links= | {{Multimedia|Links= | ||
− | + | http://demonstrations.wolfram.com/DotProduct/ '''Applet''': Dot product of two vectors (free CDF-Player required) | |
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− | http://demonstrations.wolfram.com/DotProduct/ '''Applet''': | ||
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}} | }} | ||
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[[Category:Article]] | [[Category:Article]] | ||
+ | [[Category:Vectors]] |
Latest revision as of 17:18, 24 November 2017
← Back: Simple arithmetic operations | Overview: Vector algebra | Next: Cross product → |
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The dot product of two vectors results in a scalar value and is defined as
where describes the angle between the two vectors which ranges from to (see figure). The dot product is denoted with a simple point between the vectors or without any sign.
Regarding the right side of the above equation, the following correlation can be noted: If you project the vector on the vector , you get the distance . As a consequence the result of the dot product can be seen as the area of a rectangle with the side legths and . The projection can also be done contrariwise (projection of vector on vector ). So that you get the distance . The multiplication of this term with leads to a rectangle with equivalent area but different aspect ratio (see figure).
Another possibility to compute the dot product is to multiply the corresponding components and sum them up:
In general the dot product of n-dimensional vectors is computed as follows:
which is nothing else than the matrix product of the transpose of the first vector with the second vector denoted in matrix algebra:
On the basis of the described relations it appears, that the commutative law holds:
Furthermore the following special cases can be considered, that often lead to simplifications in technical context:
Multimedial educational material
http://demonstrations.wolfram.com/DotProduct/ Applet: Dot product of two vectors (free CDF-Player required) |
Literature
- Kurt Meyberg und Peter Vachenauer, Höhere Mathematik 1: Differential- und Integralrechnung. Vektor- und Matrizenrechnung, 6. Edition (Springer Berlin Heidelberg, 2001)
- Manfred Albach, Grundlagen der Elektrotechnik 1: Erfahrungssätze, Bauelemente, Gleichstromschaltungen, 3. Edition (Pearson Studium, 2011)