Dot 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 ndimensional 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 CDFPlayer 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)