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Orbital position vector, orbital velocity vector, other orbital elements. In astrodynamics and celestial dynamics, the orbital state vectors (sometimes state vectors) of an orbit are Cartesian vectors of position and velocity that together with their time () uniquely determine the trajectory of the orbiting body in space.
If (x 1, x 2, x 3) are the Cartesian coordinates and (u 1, u 2, u 3) are the orthogonal coordinates, then = + + is the length of the coordinate vector corresponding to u i. The remaining two components of curl result from cyclic permutation of indices : 3,1,2 → 1,2,3 → 2,3,1.
The vorticity would be twice the mean angular velocity vector of those particles relative to their center of mass, oriented according to the right-hand rule. By its own definition, the vorticity vector is a solenoidal field since ∇ ⋅ ω = 0. {\displaystyle \nabla \cdot {\boldsymbol {\omega }}=0.}
Velocity is the speed in combination with the direction of motion of an object. Velocity is a fundamental concept in kinematics, the branch of classical mechanics that describes the motion of bodies. Velocity is a physical vector quantity: both magnitude and direction are needed to define it.
[9] [10] It is typically formulated as the product of a unit of measurement and a vector numerical value , often a Euclidean vector with magnitude and direction. For example, a position vector in physical space may be expressed as three Cartesian coordinates with SI unit of meters.
The position of the mass is defined by the coordinate vector r = (x, y) measured in the plane of the circle such that y is in the vertical direction. The coordinates x and y are related by the equation of the circle (,) = + =, that constrains the movement of M. This equation also provides a constraint on the velocity components,
Given a subset S of R n, a vector field is represented by a vector-valued function V: S → R n in standard Cartesian coordinates (x 1, …, x n). If each component of V is continuous, then V is a continuous vector field. It is common to focus on smooth vector fields, meaning that each component is a smooth function (differentiable any number ...
Vectors in an n-dimensional Euclidean space can be represented as coordinate vectors in a Cartesian coordinate system. The endpoint of a vector can be identified with an ordered list of n real numbers (n-tuple). These numbers are the coordinates of the endpoint of the vector, with respect to a given Cartesian coordinate system, and are ...