# Physical applications of complex numbers

1. Oct 13, 2015

### Calpalned

1. The problem statement, all variables and given/known data

2. Relevant equations
see picture above

3. The attempt at a solution
I can follow most of the steps, but not all. I got confused with finding $|\frac{dz}{dt}|$. It is easy to derive $\frac{dz}{dt}$ from $z$. Normally, I would square the two components of $dz/dt$ and take the square root to get the magnitude (velocity), but here I don't have the components. Is it a general rule that if I don't have the components of a vector, to find the magnitude, I multiply by the complex conjugate?

2. Oct 13, 2015

### Staff: Mentor

For a complex number z, $|z| = \sqrt{z \cdot \bar{z}}$

3. Oct 14, 2015

### Calpalned

What is the physical difference between $|\frac{dz}{dt}|$ and $\frac{dz}{dt}$ Could you please give a real world example (ie a car, horse, etc moving) Thank you.

4. Oct 14, 2015

### Staff: Mentor

A car's speedometer gives the magnitude of the car's velocity. For example, if you drive the car around a circle at a constant speed, the speedometer needle doesn't change. Velocity is a vector quantity, so the direction of the velocity vector is changing while the car is turning, even though the speed (|v|) is not.

5. Oct 14, 2015

### Calpalned

Thank you. I understand now that the magnitude of velocity is speed

6. Oct 15, 2015

### HallsofIvy

Note that an object moving around a circle with constant speed has norm of the velocity vector constant so its derivative is 0. But the velocity vector is NOT constant so its derivative, the acceleration vector, is not 0.