Physical applications of complex numbers

[Calpalned]

Homework Statement

Homework Equations

see picture above

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?

 

[Mark44]

Homework Statement

View attachment 90171

Homework Equations

see picture above

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?

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

 

[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.

 

[Mark44]

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.

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.

 

[Calpalned]

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.

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

 

[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.