Difference between revisions of "Dot product"

Revision as of 13:33, 15 May 2014

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The dot product of two vectors results in a scalar value and is defined as

\vec{\mathbf{a}} \cdot \vec{\mathbf{b}} = a \vec{\mathbf{e}}_{a} \cdot b\vec{\mathbf{e}}_{b} = a b \cos \alpha

where $\alpha$ describes the angle between the two vectors which ranges from $0$ to $\pi$ (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 $\vec{\mathbf{b}}$ on the vector $\vec{\mathbf{a}}$ , you get the distance $b\cos\alpha$ . As a consequence the result of the dot product can be seen as the acreage of a rectangle with the side legths $a$ and $b\cos\alpha$ . The projection can also be done contrariwise (projection of vector $\vec{\mathbf{a}}$ on the direction of vector $\vec{\mathbf{b}}$ ). So that you get the distance $a\cos\alpha$ . The multiplication of this term with $b$ leads to a rectangle with equivalent acreage but different aspect ratio (see figure).

Another possibility to compute the dot product is to multiply the corresponding components and sum them up:

\vec{\mathbf{a}} \cdot \vec{\mathbf{b}} = a b\cos\alpha = a_x b_x + a_y b_y + a_z b_z

In general the dot product of n-dimensional vectors is computed as follows:

\vec{\mathbf{a}} \cdot \vec{\mathbf{b}} = a b \cos\alpha = \sum_{i=1}^{n} a_i b_i

On the basis of the described relations it appears, that the commutative law holds:

\vec{\mathbf{a}} \cdot \vec{\mathbf{b}} = \vec{\mathbf{b}} \cdot \vec{\mathbf{a}}

Furthermore the following special cases can be considered, that often lead to simplifications in technical context:

\begin{align} \vec{\mathbf{a}} \cdot \vec{\mathbf{b}} &= ab& &\text{if}& \vec{\mathbf{a}} & \upuparrows \vec{\mathbf{b}}& &(\text{because} \cos(0) = 1)\\ \vec{\mathbf{a}} \cdot \vec{\mathbf{b}} &= 0& &\text{if}& \vec{\mathbf{a}} &\perp \vec{\mathbf{b}}& &(\text{because} \cos(\frac{\pi}{2}) = 0)\\ \vec{\mathbf{a}} \cdot \vec{\mathbf{b}} &= -ab& &\text{if}& \vec{\mathbf{a}} &\downarrow\uparrow \vec{\mathbf{b}}& &(\text{because} \cos(\pi) = -1) \end{align}

Example: Arbeit im elektrischen Feld

Ein häufiger Anwendungsfall des Skalarprodukts ergibt sich bei dem Begriff der Arbeit. Bewegt man beispielsweise eine Punktladung $Q$ in einem homogenen elektrischen Feld $\vec{\mathbf{E}}$ entlang einer geraden Strecke $\vec{\mathbf{s}}$ von einem Punkt $A$ zu einem Punkt $B$ , so lässt sich die dabei aufgewendete Arbeit mit Hilfe des Skalarprodukts wie folgt bestimmen:

W_{AB} = \vec{\mathbf{F}} \cdot \vec{\mathbf{s}}

Bei der Verschiebung der Ladung muss die Coulomb-Kraft $\vec{\mathbf{F}} = Q\vec{\mathbf{E}}$ aufgebracht werden, das heißt in diesem Fall ergibt sich der folgende Zusammenhang:

W_{AB} = Q\vec{\mathbf{E}} \cdot \vec{\mathbf{s}} = Q E \cos{\alpha}

Allgemein, also wenn die elektrische Feldstärke nicht homogen und der Weg der Verschiebung von $A$ nach $B$ keine gerade Strecke ist, gilt:

W_{AB} = \int_{A}^{B} \vec{\mathbf{F}} \cdot \mathrm{d}\vec{\mathbf{s}} = Q \int_{A}^{B} \vec{\mathbf{E}} \cdot \mathrm{d}\vec{\mathbf{s}}

Multimedial educational material

http://www.mathresource.iitb.ac.in/linear%20algebra/example7.1/index.html Applet: Skalarprodukt zweier Vektoren (engl.)

http://www.cs.brown.edu/exploratories/freeSoftware/repository/edu/brown/cs/exploratories/applets/dotProduct/dot_product_java_browser.html Applet: Skalarprodukt zweier Vektoren

http://www.mathresource.iitb.ac.in/linear%20algebra/example7.2/index.html Applet: Skalarprodukt zweier Vektoren mit der eingeschlossenen Fläche

http://demonstrations.wolfram.com/DotProduct/ Applet: Skalarprodukt zweier Vektoren (engl./ free CDF-Player erforderlich)

http://www.math.ethz.ch/~lemuren/public/exercise/linalg/LinearCombinationInR2ETHZ.html Applet: Linearkombination im zweidimensionalem Raum

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)

@@ Line 7: / Line 7: @@
 \vec{\mathbf{a}} \cdot \vec{\mathbf{b}} = a \vec{\mathbf{e}}_{a} \cdot b\vec{\mathbf{e}}_{b} = a b \cos \alpha
 </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. <br/>
+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 acreage 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 the direction of 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 acreage but different aspect ratio (see figure).

Difference between revisions of "Dot product"

Revision as of 13:33, 15 May 2014

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