Polar to Cartesian Unit Vectors in 2D

In summary, the conversation revolves around solving for the unit vectors x-hat and y-hat in terms of r-hat and phi-hat. The given equations are r-hat=cos(phi)x-hat+sin(phi)y-hat and phi-hat=cos(phi)y-hat-sin(phi)x-hat. The initial attempt results in a complicated expression for x-hat and the question arises on how to solve for alpha and beta. The solution involves computing dot products and using examples to understand the equations better.
  • #1
leahc
2
0

Homework Statement


Solve for the unit vectors x-hat and y-hat in terms of r-hat and phi-hat.


Homework Equations


r-hat=cos(phi)x-hat+sin(phi)y-hat
phi-hat=cos(phi)y-hat-sin(phi)x-hat,


The Attempt at a Solution


I have been working on this for a really long time, and I keep getting a really complicated expression for x-hat, with everything over cos(phi)^2. That seems wrong, and I can't figure out how to solve it from only those two given equations.
 
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  • #2
Let ##\hat{x} = \alpha\hat{r} + \beta\hat{\phi}##.
Compute ##\hat{x}.\hat{r}## etc.
 
  • #3
I understand that, but how do I compute α and β?
 
  • #4
Haruspex already told you how. That's a dot product in case you didn't recognize it.
 
  • #5
Where did you get those equations? Why did you choose them? Take one example of an x-y unit vector and calculate r and phi - plotting it might help. Then take a different example and do likewise. Notice anything?
 

FAQ: Polar to Cartesian Unit Vectors in 2D

What are polar to cartesian unit vectors in 2D?

Polar to cartesian unit vectors in 2D are a way of representing points in a two-dimensional coordinate system using two unit vectors, one for the x-axis and one for the y-axis. They allow for the conversion of polar coordinates (represented by an angle and distance from the origin) to cartesian coordinates (represented by x and y values).

How do you convert from polar to cartesian unit vectors in 2D?

To convert from polar to cartesian unit vectors in 2D, you can use the following formulas:

  • x = r * cos(theta)
  • y = r * sin(theta)

Where r is the distance from the origin and theta is the angle from the positive x-axis.

What are some applications of polar to cartesian unit vectors in 2D?

Polar to cartesian unit vectors in 2D are commonly used in fields such as physics, engineering, and cartography. They are especially useful in applications involving circular or rotational motion, such as in robotics or satellite navigation systems.

What is the difference between polar and cartesian coordinates?

Polar coordinates use an angle and distance from the origin to represent a point, while cartesian coordinates use x and y values. Polar coordinates are often more useful for describing circular or rotational motion, while cartesian coordinates are more commonly used in everyday applications.

Can polar to cartesian unit vectors be extended to 3D?

Yes, polar to cartesian unit vectors can be extended to 3D by adding a third unit vector for the z-axis. This allows for the representation of points in a three-dimensional coordinate system using three unit vectors. The conversion formulas are similar to those used in 2D, but with an additional z component.

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