State vector (navigation)
# 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. Mathematically, in order to describe positions in a N-dimensional space ($\mathbb {R} ^{N}$) then a state vector ${\textbf {x}}$ belongs to $\mathbb {R} ^{2N}$:

$\mathbf {x} (t)={\begin{bmatrix}x_{1}(t)\\x_{2}(t)\\x_{3}(t)\\v_{1}(t)\\v_{2}(t)\\v_{3}(t)\end{bmatrix}}$

or simply

$\mathbf {x} (t)={\begin{bmatrix}\mathbf {r} (t)\\\mathbf {v} (t)\end{bmatrix}}$

where $\mathbf {r} ={\begin{bmatrix}x_{1}&x_{2}&x_{3}\end{bmatrix}}^{\mathsf {T}}$ is the position vector and $\mathbf {v} ={\dot {\mathbf {r} }}={\begin{bmatrix}v_{1}&v_{2}&v_{3}\end{bmatrix}}^{\mathsf {T}}$ 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).

