# Position Vector

(Redirected from Displacement 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]\vec{r(t)}=r(x,y,z)[/math] or [math]\vec{r(t)}=x(t)\hat{\boldsymbol e_x}+y(t)\hat{\boldsymbol e_y}+z(t)\hat{\boldsymbol e_z}[/math] where [math](\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]\vec{r(t)}=(r,\theta,\varphi)[/math] or [math]\vec{r(t)}=r(t)\hat{\boldsymbol e_r}[/math] where [math]r[/math] is the radius, [math]\theta[/math] is inclication angle, [math]\varphi[/math] the azimuthal angle and [math](\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

- It is denoted by “

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

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

## 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*:

- In geometry, a

- [math]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/