Position Vector

A Position Vector is an Euclidean vector which corresponds distances along each axis of the reference frame from the origin to the location of a given point, particle or physical body.

• AKA: Location Vector, Radius Vector, Displacement Vector.
• Context:
  • It is denoted by “x” or “s” when it is referred to as displacement vector and “r” when it is referred to as radius vector, position vector or location vector.
  • In linear motion, displacement vector may also be defined as the shortest distance from the initial to the final position of the physical object.
  • In the Cartesian Coordinate System, it is represented as [math]\displaystyle{ \vec{r(t)}=r(x,y,z) }[/math] or [math]\displaystyle{ \vec{r(t)}=x(t)\hat{\boldsymbol e_x}+y(t)\hat{\boldsymbol e_y}+z(t)\hat{\boldsymbol e_z} }[/math] where [math]\displaystyle{ (\hat{\boldsymbol e_x},\hat{\boldsymbol e_y}, \hat{\boldsymbol e_z}) }[/math] is the basis vector.
  • In the Spherical Coordinate System, it is represented as [math]\displaystyle{ \vec{r(t)}=(r,\theta,\varphi) }[/math] or [math]\displaystyle{ \vec{r(t)}=r(t)\hat{\boldsymbol e_r} }[/math] where [math]\displaystyle{ r }[/math] is the radius, [math]\displaystyle{ \theta }[/math] is inclication angle, [math]\displaystyle{ \varphi }[/math] the azimuthal angle and [math]\displaystyle{ (\hat{\boldsymbol e_r},\hat{\boldsymbol e_\theta},\hat{\boldsymbol e_\varphi}) }[/math]is the basis vector. The relationship between the spherical and Cartesian basis vectors is given by

[math]\displaystyle{ \hat{\boldsymbol e_r} =\sin \theta \cos \varphi \,\hat{\boldsymbol e_x} + \sin \theta \sin \varphi \,\hat{\boldsymbol e_y} + \cos \theta \,\hat{\boldsymbol e_z} }[/math]

[math]\displaystyle{ \hat{\boldsymbol e_\theta} =\cos \theta \cos \varphi \,\hat{\boldsymbol e_x} + \cos \theta \sin \varphi \,\hat{\boldsymbol e_y} -\sin \theta \,\hat{\boldsymbol e_z} }[/math]

[math]\displaystyle{ \hat{\boldsymbol e_\varphi} =-\sin \varphi \,\hat{\boldsymbol e_x} + \cos \varphi \,\hat{\boldsymbol e_y} }[/math]

• Counter-Example(s):
  • Vector Norm.
  • Velocity.
  • Acceleration.
• See: Linear Motion, Euclidean Vector, Reference Frame, Cartesian Coordinate System, Spherical Coordinate System, Polar Coordinate System, Cylindrical Coordinate System.

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References

2016

• (Wikipedia, 2016) ⇒ https://www.wikiwand.com/en/Position_(vector) Retrieved:2016-5-22.
  • In geometry, a position or position vector, also known as location vector or radius vector, is a Euclidean vector that represents the position of a point P in space in relation to an arbitrary reference origin O. Usually denoted x, r, or s, it corresponds to the straight-line distances along each axis from O to P:

[math]\displaystyle{ r=\overrightarrow{OP}. }[/math]

The term "position vector" is used mostly in the fields of differential geometry, mechanics and occasionally vector calculus.

Frequently this is used in two-dimensional or three-dimensional space, but can be easily generalized to Euclidean spaces in any number of dimensions

1963

• (Feynman et al., 1963) ⇒ Richard P. Feynman, Robert B. Leighton and Matthew Sands (1963, 1977, 2006, 2010, 2013) "The Feynman Lectures on Physics": New Millennium Edition is now available online by the California Institute of Technology, Michael A. Gottlieb, and Rudolf Pfeiffer ⇒ http://www.feynmanlectures.caltech.edu/

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