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