Unit vector in polar coorindate

In summary, in rectangular coordinates, 3i+j means a length of 3 in the x-direction and 1 in the y-direction. In polar coordinates, 3r + 1\theta means a length of 3 away from the origin and an angle of 1 radian from the x-axis. To convert a vector from rectangular to polar coordinates, you can use the formulas r = |3\hat{i} + \hat{j}| = \sqrt{10} and \theta = \arctan(y/x) = \arctan(1/3). The unit vector \vec{e}_\theta represents an additional rotation from the unit vector \vec{e}_r. For more details, please
  • #1
athrun200
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In rectangular corr. 3i+j mean leght in x-direction =3 in y-3direction =1

However, how about in polar coorindate?
3r+1[itex]\theta[/itex] (r and [itex]\theta[/itex] are the unit verctor in polar coor., I don't know how to type it out, I hope you understand.)

Dose it mean a line with length 3 from origin and angle 1 radian from x-axis?
 
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  • #2
athrun200 said:
In rectangular corr. 3i+j mean leght in x-direction =3 in y-3direction =1

However, how about in polar coorindate?
3r+1[itex]\theta[/itex] (r and [itex]\theta[/itex] are the unit verctor in polar coor., I don't know how to type it out, I hope you understand.)

Dose it mean a line with length 3 from origin and angle 1 radian from x-axis?
Yes, just as 3i+j means three "steps" in the x-direction and 1 "step" in the y-direction, 3r + 1[itex]\theta[/itex] means three "steps" away from the origin and 1 "step" counter-clockwise from the x-axis.
 
  • #3
Hootenanny said:
Yes, just as 3i+j means three "steps" in the x-direction and 1 "step" in the y-direction, 3r + 1[itex]\theta[/itex] means three "steps" away from the origin and 1 "step" counter-clockwise from the x-axis.

How to convert the vector in retangular corrdinate to polar coordinate?
For example, 3i+j, how to write in polar corrdinate unit vector?
 
  • #4
For a detailed explanation see here: http://en.wikipedia.org/wiki/Polar_coordinate_system

In polar coordinates, the radius is simply the distance from the origin (i.e. the magnitude of your position vector). So in your example, [itex]r = |3\hat{i} + \hat{j}| = \sqrt{10}[/itex]. The angular component is simply the angle between your vector and the positive x-direction. Since your point is in the first quadrant, this is simply [itex]\theta=\arctan(y/x) = \arctan(1/3)[/itex]. If your point were to be in a different quadrant, you would have to adjust accordingly.
 
  • #5
I am still very unfamiliar with this new topic, and I still have a lot of question marks in my head.

The details of my reply please refer to my attachment
 

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  • #6
athrun200 said:
I am still very unfamiliar with this new topic, and I still have a lot of question marks in my head.

The details of my reply please refer to my attachment
The unit vector [itex]\vex{e}_\theta[/itex] represents an "additional" rotation from the vector [itex]\vec{e}_r[/itex]. Notice that the unit vector [itex]\vec{e}_r[/itex] is the direction from the origin to the point in space [itex]\vec{r}[/itex]: http://en.wikipedia.org/wiki/File:Polar_unit_vectors.PNG.
 

1. What is a unit vector in polar coordinate?

A unit vector in polar coordinate is a vector with a magnitude of 1 and a direction determined by an angle (θ) measured counterclockwise from the positive x-axis. It is commonly used to represent a direction or orientation in a two-dimensional space.

2. How do you find the unit vector in polar coordinate?

To find the unit vector in polar coordinate, we first calculate the magnitude of the vector using the Pythagorean theorem (r = √(x² + y²)). Then, we divide each of the components (x and y) by the magnitude to get the unit vector.

3. What is the difference between polar and Cartesian unit vectors?

The main difference between polar and Cartesian unit vectors is the way they represent direction. While polar unit vectors use an angle (θ) to determine direction, Cartesian unit vectors use the x and y components of the vector to specify direction.

4. Can a unit vector in polar coordinate have a negative magnitude?

No, a unit vector by definition has a magnitude of 1. In polar coordinate, the magnitude of a vector is always positive, as it represents the distance from the origin to the tip of the vector.

5. What is the purpose of using unit vectors in polar coordinate?

Unit vectors in polar coordinate are useful in representing direction or orientation in a two-dimensional space. They also simplify vector operations as they eliminate the need to consider the magnitude of the vector, only the direction. Additionally, they can be used to express complex numbers in the form of r(cosθ + isinθ), where r is the magnitude and θ is the angle of the vector.

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