Earth: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.
Mathematical representation
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, in order to describe positions in a N-dimensional space ([math]\displaystyle{ \mathbb{R}^N }[/math]) then a state vector [math]\displaystyle{ \textbf{x} }[/math] belongs to [math]\displaystyle{ \mathbb{R}^{2N} }[/math]:
[math]\displaystyle{ \mathbf{x}(t) = \begin{bmatrix} x_1(t)\\ x_2 (t)\\ x_3(t) \\ v_1(t) \\ v_2 (t) \\ v_3 (t) \end{bmatrix} }[/math]
or simply
[math]\displaystyle{ \mathbf{x}(t) = \begin{bmatrix} \mathbf{r}(t) \\ \mathbf{v}(t)\end{bmatrix} }[/math]
where [math]\displaystyle{ \mathbf{r} = \begin{bmatrix} x_1 & x_2 & x_3 \end{bmatrix}^\mathsf{T} }[/math] is the position vector and [math]\displaystyle{ \mathbf{v} = \dot{\mathbf{r}} = \begin{bmatrix} v_1 & v_2 & v_3 \end{bmatrix}^\mathsf{T} }[/math] is the velocity vector.
Since there is freedom to choose 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).
See also
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