Earth: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 $\text{x}$ 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

$\mathbf{𝐱}(t)=(\genfrac{}{}{0ex}{}{\mathbf{𝐫}(t)}{\mathbf{𝐯}(t)})$

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).