Rectangular coordinates

Rectangular coordinates (also Cartesian coordinates) are our typical $x,y,z$ coordinates. We express them with unit vectors $\hat{i},\hat{j},\hat{k}$ for one unit in each direction. For bodies in motion, this is not very useful, and we instead turn to polar and normal-tangential systems for expressing movement.

We also note that the angle $\text{v}$ makes with the positive $x$-axis can be determined with:

$$\frac{dy}{dx}=\tan \theta =\frac{v_y}{v_x}$$

Kinematics

Rectangular coordinates are useful when the $x$ and $y$-components of acceleration are independently generated or determined — for instance, projectile motion. If we express $r$ as the displacement from the origin, then we have the following equations.

$$\text{r}=x\text{i}+y\text{j}$$ $$\text{v}=\dot{\text{r}}=\dot{x}\text{i}+\dot{y}\text{j}$$ $$\text{a}=\dot{\text{v}}=\ddot{\text{r}}=\ddot{x}\text{i}+\ddot{y}\text{j}$$

We can also express velocity and acceleration as scalars, should we take their magnitude.

Conversion

To convert from polar to rectangular coordinates, we can use a transformation matrix:

$$\text{M}=\left[\begin{array}{ccc}{\hat{r}}_x& & {\hat{\theta }}_x\\ {\hat{r}}_y& & {\hat{\theta }}_y\end{array}\right]$$