I Coordinate System: Understanding Polar Vectors

[Devil Moo]

Hello!

I understand the the polar coordinate system without vectors. But when it is related to vector, it is confusing. Do the unit vectors r and phi keep changing?
How do I interpret it as they changes? For example, F = 2 r + 3 phi. Based on the vector addition and scale multiplication, it shall put 3 unit vectors phi to the head of 2 unit vectors r. But it seems it is wrong.

 

[andrewkirk]

Do the unit vectors r and phi keep changing?

Yes, the direction in which these two vectors point varies according to where they are situated. They form vector fields, where each point p in the number plane has a unit radial vector $\hat{\mathbf r}_p$ and a unit tangential vector $\hat{\mathbf \theta}_p$.

For Cartesian coordinates the unit vectors also form a vector field, but they point in the same direction regardless of where they are situated, so one is able to ignore the dependence on their location.

 

[Devil Moo]

For Cartesian coordinates the unit vectors also form a vector field, but they point in the same direction regardless of where they are situated, so one is able to ignore the dependence on their location.

If the unit vectors are changing, it is not appropriate to directly use the vector addition and scalar multiplication because of the different rules. Suppose F = 2r and E = 3r + pi/2 * phi. In Cartesian Coordinate, F + E = x, whereas F + E = 5r + pi/2 * phi in Polar Coordinate. How do work with that?

 

[andrewkirk]

You can only meaningfully add vectors that are based at the same point. What that means for you depends on the context. There's not enough information in your posts to allow somebody else to understand the context.

 

[Devil Moo]

Suppose F = 2r and E = 3r + pi/2 * phi. Transform from Polar to Cartesian Coordinate, F = 2x and E = -3x.
How about the resultant vector F + E? By vector addition, F + E = -x in Cartesian Coordinate.
What is the answer in Polar Coordinate? It seems it is not appropriate to use the same technique, vector addition.
Do we have to transform to Cartesian then calculate and finally transform back to Polar Coordinate?

 

[FactChecker]

Polar coordinates are most interesting where vector directions and lengths are being considered. It is unusual to directly add a direction and a length as in
F = 2 r + 3 phi.
In fact, one of the main reasons for using coordinates at right angles (orthogonal coordinates) is to be able to keep them separate. Changing one does not affect the other. So vectors can be rotated or magnified as separate independent operations without worrying about an intrinsic (coordinate-forced) relationship between the two.
If I want to rotate the vector (direction_degrees, length_ft) = (10, 4) by 5 degrees, it is (15, 4). That would be much harder in Cartesian coordinates.

 

[Stephen Tashi]

Hello!
I understand the the polar coordinate system without vectors. But when it is related to vector, it is confusing. Do the unit vectors r and phi keep changing?

You have to distinguish between "vectors" and "coordinates". Not all coordinate systems are implemented by defining the coordinates of a thing to be coefficients involved in expressing that thing as sum of vectors. You are correct in thinking that in polar coordinate system the coordinates $(r,\phi)$ do not represent the vector defined by the sum $r e_r + \phi e_{\phi}$ where $e_r$ and $e_{\phi}$ are unit vectors that remained fixed for all the vectors that we represent in polar coordinates.

"Coordinates" are more general concept than vectors. For example you might "coordinatize" a persons office location by giving two numbers ( F, R) where F is the floor level and R is the room number. This doesn't mean that there is such a thing as a "unit room number vector" that points in certain direction. As another example, a study might coordinatize a person's condition by describing it with a triple of numbers ( age, height, weight). This does not imply that a person's condition is a vector. If you wanted these conditions to be a vector, you'd have to find an interpretation where multiplying by a scalar such as (-1) also described such a condition.

Where you often see polar coordinates seeming to be interpreted as vectors is in physics where both coordinates $r$ and $\phi$ are changing with time. Such situations are often analyzed using what I would call a "moving coordinate system". If you look at what is happening "at time $t$" and the changes that occur in a small time $dt$ then the small change $dr$ in the current radial direction and the small change $d\phi$ in the angle can be considered as cartesian coordinates for a vector in a coordinate system where the "unit radial vector" points along the current radius and the "unit angle vector" is perpendicular to it.

Analyses written in symbols such as "$r$", "$dr$", "$\phi$", "$d\phi$", "$t$" look the same as all times, as long as we don't substitute a particular value of time $t$ in the expressions. But such analyses don't imply that there is a particular "unit radial vector" that stays constant over all times.

 

[Devil Moo]

So in Cartesian Coordinates System and Polar Coordinates System, the "rules" to express same vector is different. It is not appropriate to interpret the vector in Polar form as one in Cartesian.

How do we define mathematical operations in Polar form, for example, differentiation? Is it by building equations between Polar and Cartesian?

 

[chiro]

The basis doesn't change but the coefficients do.

When you have a linear system you write a vector as a linear function of basis vectors.

The basis itself stays the same and you find a way to combine the basis elements to get something else.

It's a lot like chemical elements - the constituents of "protons" and "electrons" don't change but their structure of how much there are does.