Editing Euclidean vector (section)

===Length===<!-- This section is linked from [[Law of cosines]] -->
The ''[[length]]'',  ''[[Magnitude (mathematics)|magnitude]]'' or ''[[Norm (mathematics)|norm]]'' of the vector '''a''' is denoted by ‖'''a'''‖ or, less commonly, |'''a'''|, which is not to be confused with the [[absolute value]] (a scalar "norm").

The length of the vector '''a''' can be computed with the ''[[Euclidean norm]]'',

<math display=block>\left\|\mathbf{a}\right\|=\sqrt{a_1^2+a_2^2+a_3^2},</math>

which is a consequence of the [[Pythagorean theorem]] since the basis vectors '''e'''<sub>1</sub>, '''e'''<sub>2</sub>, '''e'''<sub>3</sub> are orthogonal unit vectors.

This happens to be equal to the square root of the [[dot product]], discussed below, of the vector with itself:

<math display=block>\left\|\mathbf{a}\right\|=\sqrt{\mathbf{a}\cdot\mathbf{a}}.</math>

====Unit vector====
[[Image:Vector normalization.svg|class=skin-invert-image|thumb|right|The normalization of a vector '''a''' into a unit vector '''â''']]
{{main|Unit vector}}

A ''unit vector'' is any vector with a length of one; normally unit vectors are used simply to indicate direction. A vector of arbitrary length can be divided by its length to create a unit vector.<ref name="1.1: Vectors"/> This is known as ''normalizing'' a vector. A unit vector is often indicated with a hat as in '''â'''.

To normalize a vector {{nowrap|1='''a''' = (''a''<sub>1</sub>, ''a''<sub>2</sub>, ''a''<sub>3</sub>)}}, scale the vector by the reciprocal of its length ‖'''a'''‖. That is:

<math display=block>\mathbf{\hat{a}} = \frac{\mathbf{a}}{\left\|\mathbf{a}\right\|} = \frac{a_1}{\left\|\mathbf{a}\right\|}\mathbf{e}_1 + \frac{a_2}{\left\|\mathbf{a}\right\|}\mathbf{e}_2 + \frac{a_3}{\left\|\mathbf{a}\right\|}\mathbf{e}_3</math>

====Zero vector====
{{main|Zero vector}}

The ''zero vector'' is the vector with length zero. Written out in coordinates, the vector is {{nowrap|(0, 0, 0)}}, and it is commonly denoted <math>\vec{0}</math>, '''0''', or simply 0. Unlike any other vector, it has an arbitrary or indeterminate direction, and cannot be normalized (that is, there is no unit vector that is a multiple of the zero vector). The sum of the zero vector with any vector '''a''' is '''a''' (that is, {{nowrap|1='''0''' + '''a''' = '''a'''}}).