# dot product

Let $u=(u_{1},u_{2},\ldots,u_{n})$ and $v=(v_{1},v_{2},\ldots,v_{n})$ two vectors on $k^{n}$ where $k$ is a field (like $\mathbb{R}$ or $\mathbb{C}$). Then we define the dot product of the two vectors as:

 $u\cdot v=u_{1}v_{1}+u_{2}v_{2}+\cdots+u_{n}v_{n}.$

Notice that $u\cdot v$ is NOT a vector but a scalar (an element from the field $k$).

If $u,v$ are vectors in $\mathbb{R}^{n}$ and $\vartheta$ is the angle between them, then we also have

 $u\cdot v=\|u\|\|v\|\cos\vartheta.$

Thus, in this case, $u\perp v$ if and only if $u\cdot v=0$.

The special case  $u\cdot u$  of scalar product is the scalar square of the vector $u$.  In $\mathbb{R}^{n}$ it equals to the square of the length of $u$:

 $u\cdot u=\|u\|^{2}$
 Title dot product Canonical name DotProduct Date of creation 2013-03-22 11:46:33 Last modified on 2013-03-22 11:46:33 Owner drini (3) Last modified by drini (3) Numerical id 13 Author drini (3) Entry type Definition Classification msc 83C05 Classification msc 15A63 Classification msc 14-02 Classification msc 14-01 Synonym scalar product Related topic CauchySchwarzInequality Related topic CrossProduct Related topic Vector Related topic DyadProduct Related topic InvariantScalarProduct Related topic AngleBetweenLineAndPlane Related topic TripleScalarProduct Related topic ProvingThalesTheoremWithVectors Defines scalar square