# State vector (navigation)

In navigation, a 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 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 ($\mathbb {R} ^{N}$ ) then a state vector ${\textbf {x}}$ belongs to $\mathbb {R} ^{2N}$ :

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

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