Vector Operations in Polar Coordinates?

In summary, when working in \mathbb{R}^3, rectangular coordinates are more convenient for vector operations because they allow for the use of basis-vectors and component form. Using polar or spherical coordinates would not allow for this and would result in the loss of much of the reason for using vectors. However, in certain situations such as with spherically or cylindrically symmetrical vector fields, using polar coordinates may be more convenient.
  • #1
Poweranimals
68
0
Do you think you could do vector operations in polar coordinates?
 
Physics news on Phys.org
  • #2
When you're in [itex]\mathbb{R}^3[/itex] and you want to designate the tip of a vector by giving three coordinates, then you can use spherical coordinates (or polar coordinates in [itex]\mathbb{R}^2[/itex]) or any other coordinate system.

But using rectangular coordinates is much more convenient, because the vector notation in component form will mean the first component times the first basis-vector, the second component times the second basis-vector and so on. This can't be done in polar/spherical coordinates.
Also, adding vectors component-wise, and things like the dot-product are of no use either.

So much of the reason why you use vectors will be lost when going to spherical/polar coordinates.

In short, with vectors: use a linear coordinate system.
 
  • #3
Depends what you're doing. In physics, it is often convenient to use spherical polar coordinates for vector fields, particular if the field is spherically symmetrical. If you have cylindrical symmetry, cylindrical polar coordinates are often useful.

You can certainly write grad f, div V, and curl V in terms of their polar components.
 

1. What are polar coordinates and how are they used in vector operations?

Polar coordinates are a way of representing points in a two-dimensional space using a distance from the origin (known as the magnitude or radius) and an angle from a fixed reference line (known as the direction or angle). They are used in vector operations to describe the magnitude and direction of a vector.

2. How do you convert between polar and Cartesian coordinates?

To convert from polar coordinates (r, θ) to Cartesian coordinates (x, y), you can use the following equations:
x = r cos(θ)
y = r sin(θ)
To convert from Cartesian coordinates to polar coordinates, you can use the following equations:
r = √(x^2 + y^2)
θ = tan^-1(y/x)

3. What are the basic vector operations in polar coordinates?

The basic vector operations in polar coordinates are addition, subtraction, and multiplication by a scalar. They can be performed by converting the vectors to Cartesian coordinates, performing the operation, and then converting back to polar coordinates.

4. How do you find the magnitude and direction of a vector in polar coordinates?

To find the magnitude of a vector in polar coordinates, you can use the Pythagorean theorem:
||v|| = √(r^2 + θ^2)
To find the direction of a vector in polar coordinates, you can use the inverse tangent function:
θ = tan^-1(θ/r)

5. Can vector operations be performed directly in polar coordinates?

Yes, vector operations can be performed directly in polar coordinates by using the polar form of complex numbers. In this form, a vector is represented as a magnitude and an angle, and operations such as addition and multiplication can be performed using trigonometric identities.

Similar threads

  • Linear and Abstract Algebra
Replies
6
Views
875
  • Linear and Abstract Algebra
Replies
3
Views
183
  • Linear and Abstract Algebra
Replies
3
Views
1K
  • Linear and Abstract Algebra
Replies
14
Views
497
  • Linear and Abstract Algebra
Replies
6
Views
985
  • Linear and Abstract Algebra
Replies
8
Views
1K
  • Linear and Abstract Algebra
Replies
1
Views
748
Replies
1
Views
300
  • Linear and Abstract Algebra
Replies
7
Views
882
  • Linear and Abstract Algebra
2
Replies
41
Views
2K
Back
Top