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

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

${\displaystyle {\vec {s}}=x{\vec {i}}+y{\vec {j}}+z{\vec {k}}\,\!}$

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

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

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

Rotating Coordinate Frame