# State vector (geographical)

A geographical state vector is a set of data describing exactly where an object is located in space, and how it is moving. From a state vector, and sufficient mathematical conditions (e.g. the Picard-Lindelöf theorem), the object's past and future position can be determined.[citation needed]

A geographical state vector typically will contain seven elements: three position coordinates, three velocity terms, and the time at which these values were valid.[citation needed] Mathematically, if we are to describe positions in a N-dimensional space (${\displaystyle \mathbb {R} ^{N}}$) then a state vector ${\displaystyle {\textbf {x}}}$ belongs to ${\displaystyle \mathbb {R} ^{2N}}$:

${\displaystyle \mathbf {x} (t)=(x_{1}(t)\;\;x_{2}(t)\;\;x_{3}(t)\;\;v_{1}(t)\;\;v_{2}(t)\;\;v_{3}(t))^{T}}$

or simply

${\displaystyle \mathbf {x} (t)={\binom {\mathbf {r} (t)}{\mathbf {v} (t)}}}$

where ${\displaystyle \mathbf {r} =(x_{1}\;x_{2}\;x_{3})^{T}}$ is the position vector and ${\displaystyle \mathbf {v} ={\dot {\mathbf {r} }}=(v_{1}\;v_{2}\;v_{3})^{T}}$ is the velocity vector.

Due to the freedom one has in choosing coordinate systems for position, a state vector may also be expressed in a variety of coordinate systems (e.g. the North east down coordinate system).