Editing Euclidean vector (section)

==Physics==
{{main|Vector quantity}}
Vectors have many uses in physics and other sciences.

===Length and units===
In abstract vector spaces, the length of the arrow depends on a [[Dimensionless number|dimensionless]] [[Scale (measurement)|scale]]. If it represents, for example, a force, the "scale" is of [[Dimensional analysis|physical dimension]] length/force. Thus there is typically consistency in scale among quantities of the same dimension, but otherwise scale ratios may vary; for example, if "1 newton" and "5 m" are both represented with an arrow of 2&nbsp;cm, the scales are 1 m:50 N and 1:250 respectively. Equal length of vectors of different dimension has no particular significance unless there is some [[proportionality constant]] inherent in the system that the diagram represents. Also length of a unit vector (of dimension length, not length/force, etc.) has no coordinate-system-invariant significance.

===Vector-valued functions===
{{main|Vector-valued function}}
Often in areas of physics and mathematics, a vector evolves in time, meaning that it depends on a time parameter ''t''. For instance, if '''r''' represents the position vector of a particle, then '''r'''(''t'') gives a [[parametric equation|parametric]] representation of the trajectory of the particle. Vector-valued functions can be [[derivative|differentiated]] and [[integral|integrated]] by differentiating or integrating the components of the vector, and many of the familiar rules from [[calculus]] continue to hold for the derivative and integral of vector-valued functions.

===Position, velocity and acceleration===
The position of a point '''x''' = (''x''<sub>1</sub>, ''x''<sub>2</sub>, ''x''<sub>3</sub>) in three-dimensional space can be represented as a [[position vector]] whose base point is the origin
<math display=block>{\mathbf x} = x_1 {\mathbf e}_1 + x_2{\mathbf e}_2 + x_3{\mathbf e}_3.</math>
The position vector has dimensions of [[length]].

Given two points '''x''' = (''x''<sub>1</sub>, ''x''<sub>2</sub>, ''x''<sub>3</sub>), '''y''' = (''y''<sub>1</sub>, ''y''<sub>2</sub>, ''y''<sub>3</sub>) their [[Displacement (vector)|displacement]] is a vector
<math display=block>{\mathbf y}-{\mathbf x}=(y_1-x_1){\mathbf e}_1 + (y_2-x_2){\mathbf e}_2 + (y_3-x_3){\mathbf e}_3.</math>
which specifies the position of ''y'' relative to ''x''. The length of this vector gives the straight-line distance from ''x'' to ''y''. Displacement has the dimensions of length.

The [[velocity]] '''v''' of a point or particle is a vector, its length gives the [[speed]]. For constant velocity the position at time ''t'' will be
<math display=block>{\mathbf x}_t= t {\mathbf v} + {\mathbf x}_0,</math>
where '''x'''<sub>0</sub> is the position at time ''t'' = 0. Velocity is the [[#Ordinary derivative|time derivative]] of position. Its dimensions are length/time.

[[Acceleration]] '''a''' of a point is vector which is the [[#Ordinary derivative|time derivative]] of velocity. Its dimensions are length/time<sup>2</sup>.

===Force, energy, work===
[[Force]] is a vector with dimensions of mass×length/time<sup>2</sup> (N m s <sup>−2</sup>) and [[Newton's second law]] is the scalar multiplication
<math display=block>{\mathbf F} = m{\mathbf a}</math>

Work is the dot product of [[force]] and [[displacement (vector)|displacement]]
<math display=block>W = {\mathbf F} \cdot ({\mathbf x}_2 - {\mathbf x}_1).</math>
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In physics, scalars may also have a unit of measurement associated with them. For instance, [[Newton's second law]] is
:<math>{\mathbf F} = m{\mathbf a}</math>
where '''F''' has units of force, '''a''' has units of acceleration, and the scalar ''m'' has units of mass. In one possible physical interpretation of the above diagram, the scale of acceleration is, for instance, 2 m/s<sup>2</sup> : cm, and that of force 5 N : cm. Thus a scale ratio of 2.5 kg : 1 is used for mass. Similarly, if displacement has a scale of 1:1000 and velocity of 0.2 cm : 1 m/s, or equivalently, 2 ms : 1, a scale ratio of 0.5 : s is used for time.
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