Kinematics/3D Coordinate Systems

= Fixed Rectangular Coordinate Frame =

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 =