Difference between revisions of "Dot product"
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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 | + | 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: | Another possibility to compute the dot product is to multiply the corresponding components and sum them up: |
Revision as of 14:00, 15 May 2014
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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:
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://www.mathresource.iitb.ac.in/linear%20algebra/example7.1/index.html Applet: Dot product of two vectors http://www.cs.brown.edu/exploratories/freeSoftware/repository/edu/brown/cs/exploratories/applets/dotProduct/dot_product_java_browser.html Applet: Dot product of two vectors http://www.mathresource.iitb.ac.in/linear%20algebra/example7.2/index.html Applet: Dot product of two vectors with the enclosed area http://demonstrations.wolfram.com/DotProduct/ Applet: Dot product of two vectors (free CDF-Player required) http://www.math.ethz.ch/~lemuren/public/exercise/linalg/LinearCombinationInR2ETHZ.html Applet: Linear combination in two-dimensional space |
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)