Editing Vector space (section)

===Normed vector spaces and inner product spaces===
{{Main|Normed vector space|Inner product space}}
"Measuring" vectors is done by specifying a [[norm (mathematics)|norm]], a datum which measures lengths of vectors, or by an [[inner product]], which measures angles between vectors. Norms and inner products are denoted <math>| \mathbf v|</math> and {{nowrap|<math>\lang \mathbf v , \mathbf w \rang,</math>}} respectively. The datum of an inner product entails that lengths of vectors can be defined too, by defining the associated norm {{nowrap|<math display="inline">|\mathbf v| := \sqrt {\langle \mathbf v , \mathbf v \rangle}.</math>}} Vector spaces endowed with such data are known as ''normed vector spaces'' and ''inner product spaces'', respectively.{{sfn|Roman|2005|loc=ch. 9}}

Coordinate space <math>F^n</math> can be equipped with the standard [[dot product]]:
<math display=block>\lang \mathbf x , \mathbf y \rang = \mathbf x \cdot \mathbf y = x_1 y_1 + \cdots + x_n y_n.</math>
In <math>\mathbf{R}^2,</math> this reflects the common notion of the angle between two vectors <math>\mathbf{x}</math> and <math>\mathbf{y},</math> by the [[law of cosines]]:
<math display=block>\mathbf x \cdot \mathbf y = \cos\left(\angle (\mathbf x, \mathbf y)\right) \cdot |\mathbf x| \cdot |\mathbf y|.</math>
Because of this, two vectors satisfying <math>\lang \mathbf x , \mathbf y \rang = 0</math> are called [[orthogonal]]. An important variant of the standard dot product is used in [[Minkowski space]]: <math>\mathbf{R}^4</math> endowed with the Lorentz product{{sfn|Naber|2003|loc=ch. 1.2}}
<math display=block>\lang \mathbf x | \mathbf y \rang = x_1 y_1 + x_2 y_2 + x_3 y_3 - x_4 y_4.</math>
In contrast to the standard dot product, it is not [[positive definite bilinear form|positive definite]]: <math>\lang \mathbf x | \mathbf x \rang</math> also takes negative values, for example, for <math>\mathbf x = (0, 0, 0, 1).</math> Singling out the fourth coordinate—[[timelike|corresponding to time]], as opposed to three space-dimensions—makes it useful for the mathematical treatment of [[special relativity]]. Note that in other conventions time is often written as the first, or "zeroeth" component so that the Lorentz product is written <math display=block>\lang \mathbf x | \mathbf y \rang =  - x_0 y_0+x_1 y_1 + x_2 y_2 + x_3 y_3.</math>