dot product

Let $u=(u_1,u_2,\mathrm{\dots },u_n)$ and $v=(v_1,v_2,\mathrm{\dots },v_n)$ two vectors on $k^n$ where $k$ is a field (like $ℝ$ or $ℂ$). Then we define the dot product of the two vectors as:

$$u\cdot v=u_1{v}_1+u_2{v}_2+\mathrm{\cdots }+u_n{v}_n.$$

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

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

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

Thus, in this case, $u⟂v$ 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 ${ℝ}^n$ it equals to the square of the length of $u$:

$$u\cdot u={\parallel u\parallel }^2$$

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