Kinematics/3D Coordinate Systems

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Fixed Rectangular Coordinate Frame[edit]

In this coordinate system, vectors are expressed as an addition of vectors in the x, y, and z direction from a non-rotating origin. Usually \vec i \, \! is a unit vector in the x direction, \vec j \, \! is a unit vector in the y direction, and \vec k \, \! is a unit vector in the z direction.

The position vector, \vec s \, \! (or \vec r \, \!), the velocity vector, \vec v \, \!, and the acceleration vector, \vec a \, \! are expressed using rectangular coordinates in the following way:

\vec s = x \vec i + y \vec j + z \vec k \, \!

\vec v = \dot {s} = \dot {x} \vec {i} + \dot {y} \vec {j} + \dot {z} \vec {k} \, \!

 \vec a = \ddot {s} = \ddot {x} \vec {i} + \ddot {y} \vec {j} + \ddot {z} \vec {k} \, \!

Note:  \dot {x} = \frac{dx}{dt} ,  \ddot {x} = \frac{d^2x}{dt^2}

Rotating Coordinate Frame[edit]