# A Question about Unit Vectors of Cylindrical Coordinates

• I
• sams
In summary, the conversation discusses the definitions of the unit vector ##\hat{φ}## in cylindrical coordinates and whether it can be obtained by evaluating the cross-product of ##\hat{ρ}## and ##\hat{z}## or by using the right-hand rule. The direction of ##\hat{φ}## is typically defined as anti-clockwise when looking down the z-axis.

#### sams

Gold Member
I wrote the equations of the Nabla, the divergence, the curl, and the Laplacian operators in cylindrical coordinates ##(ρ,φ,z)##. I was wondering how to define the direction of the unit vector ##\hat{φ}##. Can we obtain ##\hat{φ}## by evaluating the cross-product of ##\hat{ρ}## and ##\hat{z}## or by using the right-hand rule?

Thank you so much for your help...

sams said:
I wrote the equations of the Nabla, the divergence, the curl, and the Laplacian operators in cylindrical coordinates ##(ρ,φ,z)##. I was wondering how to define the direction of the unit vector ##\hat{φ}##. Can we obtain ##\hat{φ}## by evaluating the cross-product of ##\hat{ρ}## and ##\hat{z}## or by using the right-hand rule?

Thank you so much for your help...

It's all here:

http://mathworld.wolfram.com/CylindricalCoordinates.html

##\phi## (or ##\theta## to mathematicians) is usually defined to be anti-clockwise looking down the z-axis.

sams
Thank you @PeroK for your help

## 1. What are unit vectors in cylindrical coordinates?

Unit vectors in cylindrical coordinates are a set of three mutually perpendicular vectors, known as r, θ, and z, that are used to describe the position and orientation of a point in three-dimensional space. They have a magnitude of 1 and are used to define the direction of each coordinate axis.

## 2. How are unit vectors in cylindrical coordinates represented?

In cylindrical coordinates, the unit vectors are typically represented as ĉr, ĉθ, and ĉz, where the caret symbol indicates a unit vector and the subscript represents the corresponding coordinate. Alternatively, they can also be represented using the standard Cartesian unit vectors, i, j, and k, along with the transformation equations for cylindrical coordinates.

## 3. How are unit vectors in cylindrical coordinates calculated?

The unit vectors in cylindrical coordinates can be calculated using the following equations:
ĉr = cos(θ) i + sin(θ) j
ĉθ = -sin(θ) i + cos(θ) j
ĉz = k

## 4. What is the relationship between unit vectors in cylindrical and Cartesian coordinates?

The unit vectors in cylindrical coordinates, ĉr, ĉθ, and ĉz, can be expressed in terms of the standard Cartesian unit vectors, i, j, and k, using the following transformation equations:
ĉr = cos(θ) i + sin(θ) j
ĉθ = -sin(θ) i + cos(θ) j
ĉz = k

## 5. How are unit vectors in cylindrical coordinates used in physics and engineering?

Unit vectors in cylindrical coordinates are commonly used in physics and engineering to describe the position, orientation, and motion of objects in three-dimensional space. They are particularly useful in situations involving cylindrical symmetry, such as in fluid dynamics, electromagnetism, and structural mechanics. They are also used in transformations between different coordinate systems and in vector calculus applications.