# Conversion vectors in cylindrical to cartesian coordinates

• ForTheGreater
In summary: Ax and Ay. In summary, The conversation discusses how to express a vector in cylindrical coordinates in terms of Cartesian coordinates. The Ax component is found using the equations that transform cylindrical unit vectors into Cartesian unit vectors and involves both the Ar and Atheta components. Both components are needed to determine the x coordinate of the vector.
ForTheGreater

## Homework Statement

It's just an example in the textbook. A vector in cylindrical coordinates.
A=arAr+aΦAΦ+azAz
to be expressed in cartesian coordinates.
Ax=A⋅ax=Arar⋅ax+AΦaΦ⋅ax

ar⋅ax=cosΦ
aΦ⋅ax=-sinΦ

Ax=ArcosΦ - AΦsinΦ

Looking at a figure of the unit vectors I get it. At the same time I just don't understand why ArcosΦ isn't enough to get the magnitude of the Ax component.

Because unit vector ##\hat{x}## has two components, one in the ##\hat{r}## direction and one in the ##\hat{\phi}## direction both of which depend on the angle ##\phi##. So when you express ##\hat{x}## as in ##A_x\hat{x}##, in terms of cylindrical unit vectors, you get two components which means you need the Pythagorean theorem to find the magnitude of ##A_x##.

If I have a point r=4, phi=pi/8 and z=z; in cylindrical coordinates.
sqrt( (r*cos(phi) )2 + (r*sin(phi) )2 ) = 4
So taking r*cos(phi) as the x component and r*sin(phi) as the y component seems to be enough to get the same point represented in both coordinate systems? What am I missing?

Suppose I gave you vector
$$\vec{A}=3\hat{a}_r-2 \hat{a}_{\phi}+4 \hat{z}$$
How would you proceed to find the Cartesian x-component of this vector? You will need the equations that transform the cylindrical unit vectors into the Cartesian unit vectors.

ForTheGreater said:

## Homework Statement

It's just an example in the textbook. A vector in cylindrical coordinates.
A=arAr+aΦAΦ+azAz
to be expressed in cartesian coordinates.
Ax=A⋅ax=Arar⋅ax+AΦaΦ⋅ax

ar⋅ax=cosΦ
aΦ⋅ax=-sinΦ

Ax=ArcosΦ - AΦsinΦ

Looking at a figure of the unit vectors I get it. At the same time I just don't understand why ArcosΦ isn't enough to get the magnitude of the Ax component.
You did this correctly. In terms of interpretation, ##A_r \cos {\phi}## is only the component of the r component of A in the x direction. You also need the component of the ##\phi## component of A in the x direction. This is your ##-A_{\phi}s\sin{\phi}##

Thank you, I think I just didn't think of that you'll need both the Ax component and the Ay component of the vector to get the x coordinate.

ForTheGreater said:
Thank you, I think I just didn't think of that you'll need both the Ax component and the Ay component of the vector to get the x coordinate.
I think you meant Ar and Atheta

## 1. What is the formula for converting cylindrical coordinates to cartesian coordinates?

The formula for converting cylindrical coordinates (ρ, φ, z) to cartesian coordinates (x, y, z) is:
x = ρ * cos(φ)
y = ρ * sin(φ)
z = z

## 2. How do I determine the values of ρ, φ, and z in a cylindrical coordinate system?

ρ represents the distance from the origin to the point in the xy-plane, φ represents the angle between the positive x-axis and the line segment connecting the origin to the point, and z represents the height or distance along the z-axis. These values can be determined by drawing a line from the origin to the point and using basic trigonometry to calculate the values.

## 3. Can you convert a point from cartesian coordinates to cylindrical coordinates?

Yes, you can convert a point from cartesian coordinates to cylindrical coordinates using the formula:
ρ = √(x² + y²)
φ = arctan(y/x)
z = z

## 4. What are the advantages of using cylindrical coordinates over cartesian coordinates?

Cylindrical coordinates can be useful for representing points or objects that have circular or cylindrical symmetry. They can also be easier to work with in certain situations, such as when calculating volumes or integrating over curved surfaces.

## 5. Are there any limitations to using cylindrical coordinates?

One limitation of using cylindrical coordinates is that they are not as intuitive for visualizing shapes or objects as cartesian coordinates. Also, not all objects can be easily represented or described using cylindrical coordinates, so cartesian coordinates may be more appropriate in those cases.

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