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.

A geographical state vector typically will contain seven elements: three position coordinates, three velocity terms, and the time at which these values were valid. 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).

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