Difference between revisions of "Dot product"
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:<math> | :<math> | ||
\vec{\mathbf{a}} \cdot \vec{\mathbf{b}} = a b \cos\alpha = \sum_{i=1}^{n} a_i b_i | \vec{\mathbf{a}} \cdot \vec{\mathbf{b}} = a b \cos\alpha = \sum_{i=1}^{n} a_i b_i | ||
+ | </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> | ||
+ | \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: | On the basis of the described relations it appears, that the commutative law holds: |
Revision as of 16:40, 20 March 2015
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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://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)