# Converting arbitrary Cartesian vector to cylindrical

• playoff
In summary, the conversation discusses a problem of converting a vector's Cartesian components to circular cylindrical coordinates. The speaker is unsure of how to approach the problem and asks for guidance. It is mentioned that if the z component is zero, there will be no transformation needed. The other person suggests recognizing it as a conversion from x,y to polar coordinates, with cylindrical coordinates being an extension of polar to 3D.
playoff
Hello PF, I have a problem to solve in the following form: Given a vector with Cartesian components, V={Vx,Vy,Vz}, find its components in circular cylindrical coordinate.

Given the actual vector components, it'd be very easy to convert. But I have no idea where to start on this. Any guide to where to start from will be much appreciated!

If the z component were zero then what kind conversion would it be?

Well, its a cylindrical coordinate, so Vz wouldn't go over any transformation, right? So if Vz=0, then after the conversion it will still be 0. I am still very lost...

I was hoping you'd recognize it as a conversion from x,y to polar coordinates. Cylindrical are an extension of polar to 3D by adding the z component.

## 1. What is a Cartesian vector?

A Cartesian vector is a mathematical representation of a quantity that has both magnitude and direction. It is commonly used in physics and engineering to describe physical quantities such as force, velocity, and acceleration.

## 2. What is a cylindrical vector?

A cylindrical vector is a mathematical representation of a quantity that has both magnitude and direction, but is described using cylindrical coordinates instead of Cartesian coordinates. In cylindrical coordinates, a vector is defined by its magnitude, angle from the z-axis, and height from the xy-plane.

## 3. Why would I need to convert a Cartesian vector to cylindrical?

Converting a Cartesian vector to cylindrical can be useful in situations where the coordinates are more naturally described using cylindrical coordinates. For example, in problems involving circular motion or cylindrical objects, it may be easier to use cylindrical coordinates to describe vectors.

## 4. How do you convert a Cartesian vector to cylindrical?

To convert a Cartesian vector to cylindrical, you can use the following equations:
r = √(x^2 + y^2)
θ = tan^-1(y/x)
z = z

Where r is the magnitude of the vector, θ is the angle from the z-axis, and z is the height from the xy-plane.

## 5. Can you convert a cylindrical vector back to Cartesian?

Yes, you can convert a cylindrical vector back to Cartesian using the following equations:
x = rcos(θ)
y = rsin(θ)
z = z

Where x, y, and z are the Cartesian coordinates and r and θ are the magnitude and angle of the cylindrical vector. These equations are essentially the reverse of the ones used to convert from Cartesian to cylindrical.

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