Appendix A

Appendix A: Review of useful mathematical operations for Kinematics

Review of useful mathematical operations for Kinematics

Appendix A provides an overview of some basic mathematical relations for students of kinematics

Complex number representation

Complex number notation provides a useful form for representing vectors in 2D. Two forms of complex numbers will be considered: 1) complex cartesian form, 2) complex polar form. As numbers, the traditional operations of addition, multiplication, and differentiation apply.
Further, these operations have physical meaning as shown in the table below.

Vector complex cartesian complex polar
$r$ $r_x + r_y*i$ $re^{i\theta}$
$r$ 0 $r=\sqrt{r_x^2+r_y^2}$
$\theta$ 0 $\theta=atan2(r_y,r_x}$
$r_x$ $r_x=rcos(\theta)$ 0
$r_y$ $r_y=rsin(\theta)$ 0
Physical Operation Mathematical Operation Preferred representation
Vector loop addition complex cartesian
pure rotation Multiplication (unit mag.) Complex polar
pure stretching / translation Multiplication (0 angle) complex polar

Complex Polar Notation - a few more tricks

Compex Cartersian

(1)
\begin{equation} r=r_x + r_y*i \end{equation}