Editing Momentum (section)

===Multiple dimensions===
[[File:Elastischer stoß 2D.gif|thumb|right|Two-dimensional elastic collision. There is no motion perpendicular to the image, so only two components are needed to represent the velocities and momenta. The two blue vectors represent velocities after the collision and add vectorially to get the initial (red) velocity.]]
Real motion has both direction and velocity and must be represented by a [[Vector (geometry)|vector]]. In a coordinate system with {{math|{{var|x}}, {{var|y}}, {{var|z}}}} axes, velocity has components {{math|{{var|v}}{{sub|{{var|x}}}}}} in the {{mvar|x}}-direction, {{math|{{var|v}}{{sub|{{var|y}}}}}} in the {{mvar|y}}-direction, {{math|{{var|v}}{{sub|{{var|z}}}}}} in the {{mvar|z}}-direction. The vector is represented by a boldface symbol:<ref name=FeynmanCh11>[https://feynmanlectures.caltech.edu/I_11.html ''The Feynman Lectures on Physics''] Vol. I Ch. 11: Vectors</ref>

<math display="block">\mathbf{v} = \left(v_x,v_y,v_z \right). </math>

Similarly, the momentum is a vector quantity and is represented by a boldface symbol:

<math display="block">\mathbf{p} = \left(p_x,p_y,p_z \right). </math>

The equations in the previous sections, work in vector form if the scalars {{math|p}} and {{math|v}} are replaced by vectors {{math|'''p'''}} and {{math|'''v'''}}. Each vector equation represents three scalar equations. For example,

<math display="block">\mathbf{p}= m \mathbf{v}</math>

represents three equations:<ref name=FeynmanCh11/>

<math display="block">\begin{align}
 p_x &= m v_x\\
 p_y &= m v_y \\
 p_z &= m v_z.
\end{align} </math>

The kinetic energy equations are exceptions to the above replacement rule. The equations are still one-dimensional, but each scalar represents the [[Magnitude (mathematics)#Euclidean vector space|magnitude of the vector]], for example,

<math display="block"> v^2 = v_x^2+v_y^2+v_z^2\,.</math>

Each vector equation represents three scalar equations. Often coordinates can be chosen so that only two components are needed, as in the figure. Each component can be obtained separately and the results combined to produce a vector result.<ref name=FeynmanCh11/>

A simple construction involving the center of mass frame can be used to show that if a stationary elastic sphere is struck by a moving sphere, the two will head off at right angles after the collision (as in the figure).<ref>{{harvnb|Rindler|1986|pp=26–27}}</ref>