Unit Vector polar in terms of cartesian

[aaj92]

Homework Statement

Prove that the unit vector r{hat} of two-dimensional polar coordinates is equal to r{hat}= x{hat}cosθ + y{hat}sinθ and find the corresponding expression for θ{hat}.

all I need is the last part... I'm just not sure what θ{hat} is? How do I go about doing this? Nothing in my book even hints at how to do this.

Homework Equations

x = r cos(theta)
y = r sin(theta)
r = sqrt(x^2 + y^2)
theta = arctan(y/x)

The Attempt at a Solution

I really just need help getting started... I've been staring at this for over an hour which I know is sad but r{hat} is significantly easier than theta{hat}.

 

[Simon Bridge]

How did you do the first bit?
What would be the analogous method for the second bit?
You are not asked to prove it, just write it down.

note:
$\text{\hat{a}} \rightarrow \hat{a}$ ... rather than a{hat}.

(welcome to PF)

 

[elfmotat]

I'm not sure if this will help you, but the general form of the transformation of a basis vector is:

$$\vec{e}_{\bar{\nu}}=\sum_{\mu=1}^n \frac{ \partial x^\mu }{ \partial x^{\bar{\nu}}}\vec{e_\mu}$$

where n is the number of dimensions (in this case two). x^μ represents the Cartesian coordinates x and y (i.e. x¹=x, x²=y). x^ν (with a bar over it - this distinguishes between Cartesian and polar coordinates) represents the polar coordinates r and θ.

What you need to do is differentiate the Cartesian coordinates x and y with respect to r and θ (i.e. dx/dr, dx/dθ, dy/dr, and dy/dθ). When you sum the Cartesian basis vectors e₁=(1,0) and e₂=(0,1) times the appropriate values, you'll get basis vectors for r and θ.

 

[kusiobache]

Could someone give me a hint on the first part of this? Because I can derive it - that is just simple trigonometry - but I can't figure out how to concretely prove that $\hat{r}= \hat{x}cosθ + \hat{y}sinθ$

Edit: I'm thinking illustrate that $\vec{r} = r\hat{r}$ in polar and then showing that in Cartesian $\vec{r} = \hat{x}cos\phi +\hat{y}sin\phi$

Edit2: Nope, I'm confused again.. I think elfmotat is correct, but I don't quite understand his explanation.

Edit3: nevermind - I got it.

 

[asteriatic]

No need to feel discouraged, sometimes problems can be challenging and it's always good to ask for help when needed. Let's break down the problem and see if we can make it more manageable.

First, let's define what θ{hat} is. In polar coordinates, θ represents the angle between the positive x-axis and the vector r. θ{hat} would then be the unit vector in the direction of this angle, which we can express as cosθ{hat} and sinθ{hat} since these are the components of the unit vector in the x and y directions.

Now, we can use the relationships given in the homework statement to express r{hat} in terms of x and y. Substituting x and y in the equation for r{hat}, we get:

r{hat} = (x/r)x{hat} + (y/r)y{hat}

Since r = √(x^2 + y^2), we can rewrite this as:

r{hat} = (x/√(x^2 + y^2))x{hat} + (y/√(x^2 + y^2))y{hat}

Now, we can use the trigonometric identities sinθ = y/r and cosθ = x/r to rewrite this as:

r{hat} = (x/r)cosθ + (y/r)sinθ

And since we know that r{hat} = x{hat}cosθ + y{hat}sinθ, we can conclude that:

θ{hat} = x{hat}sinθ - y{hat}cosθ

I hope this helps you get started on the problem. Remember to always break down complicated problems into smaller, more manageable steps and use the relationships and identities given to guide your solution. Good luck!