section_path: Definition in classical mechanics/Multiple dimensions
section_name: Multiple dimensions
paragraph_text: Multiple dimensions
  
  Real motion has both direction and velocity and must be represented by a vector. In a coordinate system with x, y, z axes, velocity has components v in the x -direction, v in the y -direction, v in the z -direction. The vector is represented by a boldface symbol:
  
  v = (v x, v y, v z). {\displaystyle \mathbf {v} =\left(v_{x},v_{y},v_{z}\right).}
  
  Similarly, the momentum is a vector quantity and is represented by a boldface symbol:
  
  p = (p x, p y, p z). {\displaystyle \mathbf {p} =\left(p_{x},p_{y},p_{z}\right).}
  
  The equations in the previous sections, work in vector form if the scalars p and v are replaced by vectors p and v. Each vector equation represents three scalar equations. For example,
  
  p = m v {\displaystyle \mathbf {p} =m\mathbf {v} }
  
  represents three equations:
  
  p x = m v x p y = m v y p z = m v z. {\displaystyle {\begin{aligned}p_{x}&=mv_{x}\\p_{y}&=mv_{y}\\p_{z}&=mv_{z}.\end{aligned}}}
  
  The kinetic energy equations are exceptions to the above replacement rule. The equations are still one-dimensional, but each scalar represents the magnitude of the vector, for example,
  
  v 2 = v x 2 + v y 2 + v z 2. {\displaystyle v^{2}=v_{x}^{2}+v_{y}^{2}+v_{z}^{2}\,.}
  
  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.
  
  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).