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 $\overrightarrow{i}$ is a unit vector in the x direction, $\overrightarrow{j}$ is a unit vector in the y direction, and $\overrightarrow{k}$ is a unit vector in the z direction.

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

$\overrightarrow{s}=x\overrightarrow{i}+y\overrightarrow{j}+z\overrightarrow{k}$

$\overrightarrow{v}=\dot{s}=\dot{x}\overrightarrow{i}+\dot{y}\overrightarrow{j}+\dot{z}\overrightarrow{k}$

$\overrightarrow{a}=\ddot{s}=\ddot{x}\overrightarrow{i}+\ddot{y}\overrightarrow{j}+\ddot{z}\overrightarrow{k}$

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