Polar Coordinates Homework: Find Equations & Solutions

[v_pino]

Homework Statement

Give the equations for the plane polar unit vectors

^ ^
r and (theta)
- -

in terms of

the Cartesian unit vectors
^ and ^ and hence show that
i j
- -

^ . ^
dr / dt = (theta) x theta
- -

Homework Equations

The Attempt at a Solution

I don't know how I can approach this question as I missed my lesson. I tried to google Polar Coordinates but the materials that I found weren't closely related to this. What topic should I look up? Any recommended websites? Thanks!

 

[kuruman]

You will find what you need here

http://en.wikipedia.org/wiki/Polar_coordinate_system

then take derivatives remembering the product and chain rules of differentiation.

 

[kerimek]

Hello, there! I understand your confusion with this question. Let me explain the topic of polar coordinates and how it relates to this question.

Polar coordinates are a way of representing points in a two-dimensional plane using a distance and an angle. The distance is represented by the letter "r" and the angle is represented by the Greek letter "theta" (θ). These coordinates are often used in mathematics and physics to describe circular or rotational motion.

In order to find the equations for the plane polar unit vectors, we need to first understand how they relate to the Cartesian unit vectors (i and j). The polar unit vector in the direction of increasing r is denoted by r^ and the unit vector in the direction of increasing θ is denoted by θ^. These unit vectors are perpendicular to each other and to the Cartesian unit vectors.

To find the equations for r^ and θ^, we can use the following relationships:

r^ = cos(θ)i + sin(θ)j
θ^ = -sin(θ)i + cos(θ)j

Using these equations, we can show that the derivative of r with respect to time (dr/dt) is equal to θ^ x r^. This is because the cross product of two vectors is equal to the magnitude of the first vector multiplied by the magnitude of the second vector, multiplied by the sine of the angle between them. In this case, the angle between θ^ and r^ is 90 degrees, so the sine of the angle is 1. This simplifies the equation to dr/dt = r^ x θ^.

I hope this explanation helps you understand the topic of polar coordinates and how it relates to this question. If you need further assistance, I would recommend looking up online resources or consulting with a teacher or tutor. Best of luck with your homework!