What is the significance of \hat{\theta} in polar vector coordinates?

In summary, the conversation centers around the use and understanding of the vector \textbf{r} and its relation to \hat{\textbf{r}} and \hat{\theta} in polar coordinates. The main point of confusion is how the vector is described by these unit vectors and their linear combination, and the difference between describing a vector in cartesian versus polar coordinates. The conversation also questions the physical meaning of the linear combination of \hat{\textbf{r}} and \hat{\theta}} and whether it is necessary to use both in describing a vector.
  • #1
ice109
1,714
6
i don't understand the point of [itex]\hat{\theta}[/itex] if a vector is completely described by [itex] \textbf{r}=r \hat{\textbf{r}}[/itex]

btw tex is doing something weird, apparently i can't make greek letters bold
[tex]\textbf{\delta}[/tex]
 
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  • #2
no one of you math geniuses can answer this for me?
 
  • #3
I don't pretend to be a math genius but perhaps none of them understands your question. What do you mean by "a vector is completely described by [itex] \textbf{r}=r \hat{\textbf{r}}[/itex]". Are you talking about a specific vector? Because that certainly does not "completely describe" a general vector. If you have a vector "completely described" by [itex] \textbf{r}=r \hat{\textbf{r}}[/itex] then you don't need [itex]\theta'[/itex].

If you have formulas for both r' and [itex]\theta'[/itex], what makes you think that the vector is "completely described" by [itex] \textbf{r}=r \hat{\textbf{r}}[/itex]
? Perhaps it would help if you stated the precise problem.
 
  • #4
[tex]\hat{\textbf{r}}[/itex] depends on [tex]\theta[/tex]... It changes according to the angle. Unless you know what [tex]\theta[/tex] is you can't draw [tex]\hat{\textbf{r}}[/itex]
 
  • #5
HallsofIvy said:
I don't pretend to be a math genius but perhaps none of them understands your question. What do you mean by "a vector is completely described by [itex] \textbf{r}=r \hat{\textbf{r}}[/itex]". Are you talking about a specific vector? Because that certainly does not "completely describe" a general vector. If you have a vector "completely described" by [itex] \textbf{r}=r \hat{\textbf{r}}[/itex] then you don't need [itex]\theta'[/itex].

If you have formulas for both r' and [itex]\theta'[/itex], what makes you think that the vector is "completely described" by [itex] \textbf{r}=r \hat{\textbf{r}}[/itex]
? Perhaps it would help if you stated the precise problem.

does [itex]\textbf{r}[/itex] describe a general vector in cartesian coordinates? if it does then i don't see any difference between the position vector in cartesian coordinates and in polar coordinates.

in fact i don't even understand the physical meaning of a linear combination of [itex]\hat{\textbf{r}}[/itex] and [itex]\hat{\theta}[/itex]. actually that is erroneous , i have no problem visualizing the resultant of these two vectors, i would just need to connect them head to tail. what i don't understand is what i said before, what is the point of the [itex]\hat{\theta}}[/itex]

the picture represents my understanding of the the polar coordinates in terms of the cartesian coordinates where [itex]\textbf{A}[/itex] is the vector I'm trying to describe in terms of the the polar unit vectors. is it correct? and if it is correct why can't describe [itex]\textbf{A}[/itex] by just scaling the [itex]\hat{\textbf{r}}[/itex] a little and making its [itex]\theta[/itex] argument little bigger?
 

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What are polar vector coordinates?

Polar vector coordinates are a way of representing a vector in two-dimensional space using magnitude and direction. They are often used in physics and engineering to describe the movement of objects.

How do you convert polar vector coordinates to Cartesian coordinates?

To convert polar vector coordinates to Cartesian coordinates, you can use the following formulas:
x = r * cos(theta)
y = r * sin(theta)
Where r is the magnitude of the vector and theta is the direction.

What is the difference between a polar vector and a polar coordinate?

A polar vector and a polar coordinate both use the same concept of magnitude and direction, but a polar vector represents a physical quantity with magnitude and direction, while a polar coordinate represents a point in space with a certain distance and direction from the origin.

How do polar vector coordinates relate to polar coordinates?

Polar vector coordinates are a specific type of polar coordinates that are used to describe vectors. They are often used in situations where the direction and magnitude of a vector are important.

What are some real-life applications of polar vector coordinates?

Polar vector coordinates are commonly used in navigation systems such as GPS, as well as in physics and engineering to describe the movement of objects. They are also used in aviation for flight planning and control.

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