Velocity
Cartesian coordinates

In multi-dimensional Cartesian coordinate systems, velocity is broken up into components that correspond with each dimensional axis of the coordinate system. In a two-dimensional system, where there is an x-axis and a y-axis, corresponding velocity components are defined as

v x = d x / d t, {\displaystyle v_{x}=dx/dt,}

v y = d y / d t. {\displaystyle v_{y}=dy/dt.}

The two-dimensional velocity vector is then defined as v =< v x, v y > {\displaystyle {\textbf {v}}=<v_{x},v_{y}>}. The magnitude of this vector represents speed and is found by the distance formula as

| v | = v x 2 + v y 2. {\displaystyle |v|={\sqrt {v_{x}^{2}+v_{y}^{2}}}.}

In three-dimensional systems where there is an additional z-axis, the corresponding velocity component is defined as

v z = d z / d t. {\displaystyle v_{z}=dz/dt.}

The three-dimensional velocity vector is defined as v =< v x, v y, v z > {\displaystyle {\textbf {v}}=<v_{x},v_{y},v_{z}>} with its magnitude also representing speed and being determined by

| v | = v x 2 + v y 2 + v z 2. {\displaystyle |v|={\sqrt {v_{x}^{2}+v_{y}^{2}+v_{z}^{2}}}.}

While some textbooks use subscript notation to define Cartesian components of velocity, others use u {\displaystyle u}, v {\displaystyle v}, and w {\displaystyle w} for the x {\displaystyle x} -, y {\displaystyle y} -, and z {\displaystyle z} -axes respectively.
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