# Vector transformation, cylindrical to Cartesian

• tomwilliam
In summary, the goal is to convert a result given in cylindrical coordinates to cartesian coordinates. The unit vectors and equation for r are provided, but the conversion process gets stuck. However, by using trigonometric identities and simplifying, the final answer is found to be k times the cross product of the unit vectors e_x and e_y.
tomwilliam

## Homework Statement

I have a result which is in the form (cylindrical coordinates):

$$A\boldsymbol{e_{\theta }}=kr\boldsymbol{e_{\theta }}$$

And I have to provide the answer in cartesian coordinates.

## Homework Equations

I know that the unit vectors:

$$\boldsymbol{\hat{\theta} }=\begin{bmatrix}-sin\ \theta \\ cos\ \theta \end{bmatrix}$$

and that
$$r=\sqrt{x^{2}+y^{2}}$$

## The Attempt at a Solution

$$kr\boldsymbol{e_{\theta }} =k\left (\sqrt{x^{2}+y^{2}}\right )$$

$$\left (-sin\left(tan^{-1}\left(\frac{y}{x}\right )$$
$$\right )\boldsymbol{e_{x}}$$
$$+cos\left (tan^{-1}\left (\frac{y}{x}\right )\boldsymbol{e_{y}} \right )\\$$

I can't seem to get further than this. I don't know if I've made a mistake, or whether there is some trig identity that can help me simplify further, but I know the final answer and it is much simpler.
Any help much appreciated.
P.S. Why does the latex break down when the equation is too long?

Last edited:
Is there a way to fix this? A:$$kr\boldsymbol{e_{\theta }}=k\left (\sqrt{x^{2}+y^{2}}\right )\begin{bmatrix}-sin\ \theta \\ cos\ \theta \end{bmatrix}$$$$=k\left (\sqrt{x^{2}+y^{2}}\right )\begin{bmatrix}-\sin\left (\tan ^{-1}\left (\frac{y}{x} \right ) \right ) \\ \cos\left (\tan ^{-1}\left (\frac{y}{x} \right ) \right ) \end{bmatrix}$$$$=k\begin{bmatrix}-x\sin\left (\tan ^{-1}\left (\frac{y}{x} \right ) \right ) \\ y\cos\left (\tan ^{-1}\left (\frac{y}{x} \right ) \right ) \end{bmatrix}$$$$=k\begin{bmatrix}-x\frac{y}{\sqrt{x^2+y^2}}\\ y\frac{x}{\sqrt{x^2+y^2}}\end{bmatrix}$$$$=k\begin{bmatrix}-y\\ x\end{bmatrix}$$$$=k\boldsymbol{e_{x}}\times \boldsymbol{e_{y}}$$

## 1. What is vector transformation from cylindrical to Cartesian coordinates?

Vector transformation from cylindrical to Cartesian coordinates is the process of converting a vector expressed in cylindrical coordinates (r, θ, z) to its equivalent representation in Cartesian coordinates (x, y, z). This allows for easier visualization and calculations in three-dimensional space.

## 2. How do you perform a vector transformation from cylindrical to Cartesian coordinates?

To perform a vector transformation from cylindrical to Cartesian coordinates, you can use the following equations:
x = r * cos(θ)
y = r * sin(θ)
z = z

## 3. Can you explain the difference between cylindrical and Cartesian coordinates?

Cylindrical coordinates use a distance from the origin (r), an angle from a fixed axis (θ), and a height or depth (z) to describe a point in three-dimensional space. Cartesian coordinates use three perpendicular axes (x, y, z) to describe a point in three-dimensional space.

## 4. Why is vector transformation from cylindrical to Cartesian coordinates important?

Vector transformation from cylindrical to Cartesian coordinates is important because it allows for easier visualization and calculations in three-dimensional space. It also allows for the use of standard vector operations, such as dot product and cross product, in cylindrical coordinate systems.

## 5. What is an example of a real-life application of vector transformation from cylindrical to Cartesian coordinates?

One example of a real-life application of vector transformation from cylindrical to Cartesian coordinates is in computer graphics, where objects are often defined in Cartesian coordinates, but may need to be transformed into cylindrical coordinates for certain calculations or simulations.

• Introductory Physics Homework Help
Replies
10
Views
486
• Introductory Physics Homework Help
Replies
1
Views
1K
• Introductory Physics Homework Help
Replies
11
Views
330
• Introductory Physics Homework Help
Replies
1
Views
255
• Introductory Physics Homework Help
Replies
6
Views
323
• Introductory Physics Homework Help
Replies
3
Views
2K
• Introductory Physics Homework Help
Replies
9
Views
1K
• Introductory Physics Homework Help
Replies
1
Views
249
• Introductory Physics Homework Help
Replies
7
Views
2K
• Introductory Physics Homework Help
Replies
5
Views
951