Unit vectors in polar co ordinates

In summary, the radial and traversal unit vectors are vector functions of the scalar variable θ and are used to calculate the magnitude of velocity and acceleration vectors in polar coordinates. To simplify this calculation, it is necessary to transform the basis to Cartesian coordinates, where the metric tensor is trivial. The unit vectors in the r and θ directions are defined as the directions in which r and θ are not changing, with components of (1,0) and (0,1) in the (r,θ) basis.
  • #1
manimaran1605
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0
I have two questions
1) How the radial and traversal unit vectors are vector funcitons of scalar variable θ (angle between the position vector and polar axis.
2) To find velocity and accleration in polar co ordinates why it is need to write the traversal and radial unit vectors by transforming its basis to cartesian co ordinate basis i,j? I have little bit idea that it is for our convenience that we are transforming basis to simplify the problem to get the velocity, but i don't know tell how it is convenient
 
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  • #2
manimaran1605 said:
I have two questions
1) How the radial and traversal unit vectors are vector functions of scalar variable θ (angle between the position vector and polar axis.
2) To find velocity and acceleration in polar co ordinates why it is need to write the traversal and radial unit vectors by transforming its basis to cartesian co ordinate basis i,j? I have little bit idea that it is for our convenience that we are transforming basis to simplify the problem to get the velocity, but i don't know tell how it is convenient

Answering #2 first: We're trying to calculate the magnitude of the velocity and acceleration vectors, and it's really easy to calculate the magnitude of a vector if you know its components in the Cartesian basis: ##S=\sqrt{x^2+y^2}##. If you have the components in a different basis such as polar coordinates, you can still calculate the magnitude but you have to use a thing called the "metric tensor" - you can't just use the simple Pythagorean theorem - and that's often just as much or more work than converting into Cartesian coordinates. (Actually, there's nothing special about Cartesian coordinates, it just so happens that the metric tensor in that coordinate system is so trivial that you don't notice it).

And for #1: the unit vector in the ##r## direction at a point is a unit vector in the direction that ##\theta## is not changing, and the unit vector in the ##\theta## direction is a unit vector pointing in the direction that ##r## is not changing. These vectors will point in different directions at different points in the plane, but their components in the ##(r,\theta)## basis will not; the vectors are ##(1,0)## and ##(0,1)## everywhere when using polar coordinates.
 

1. What are unit vectors in polar coordinates?

Unit vectors in polar coordinates are used to represent the direction of a point in a polar coordinate system. They have a magnitude of 1 and are used to define the direction of the radial and angular components of a point in polar coordinates.

2. How are unit vectors in polar coordinates calculated?

Unit vectors in polar coordinates are calculated by taking the cosine and sine of the angle of the point in the polar coordinate system. The cosine value represents the x-component of the unit vector, and the sine value represents the y-component.

3. Why are unit vectors in polar coordinates important?

Unit vectors in polar coordinates are important because they allow us to easily represent the direction of points in a polar coordinate system. They are also used in various mathematical calculations, such as finding the gradient and divergence of a vector field.

4. How do unit vectors in polar coordinates differ from unit vectors in Cartesian coordinates?

Unit vectors in polar coordinates and Cartesian coordinates are different in the way they represent the direction of a point. In polar coordinates, the unit vectors are oriented along the radial and angular directions, while in Cartesian coordinates, they are oriented along the x and y axes.

5. Can unit vectors in polar coordinates be converted to Cartesian coordinates?

Yes, unit vectors in polar coordinates can be converted to Cartesian coordinates by using the trigonometric relationships between the two coordinate systems. The x-component of the unit vector is equal to the cosine of the angle, and the y-component is equal to the sine of the angle.

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