Finding Polar Unit Vectors from Cartesian Vector - Pete

[petertheta]

I have a worksheet that due to missing the lecture I'm now stuck on.

You are given a cartesian vector and told find the polar unit vecors and hence express the original vector as a linear combination of the polar unit vectors just found. I've searched resources online but feel that there is conflicting information. It would be good if you could help clarify the methodology to do this transformation. I generally understand the nature of the unit vector.

So here's the question:

$$\vec{v} = 3\hat{x} + 4\hat{y}$$

Where the x-hat etc are the cartesian unit vectors.

But what I have found through reading through online notes etc gives the polar unit vectors as:
$$\hat{r} = \cos{\theta}\hat{x}+\sin{\theta}\hat{y}$$
$$\hat{\theta} = -\sin{\theta}\hat{x} + \cos{\theta}\hat{y}$$


The thing is though these are still containing the cartesian unit vectors so I can't really see how a transformation has taken place.

Can you help?

Thanks - Pete

 

[tiny-tim]

Hi Pete!

You are given a cartesian vector and told find the polar unit vecors and hence express the original vector as a linear combination of the polar unit vectors just found.

i'm not sure i understand the question

if the origin is at O = (0,0), and if P = (3,4),

then the vector OP is (3,4), and the unit polar vectors are (3/5,4/5) and (-4/5,3/5)

so OP = 5(3/5,4/5) + 0(-4/5,3/5)

but usually you are given a vector PQ, and asked to express that as a combination of (3/5,4/5) and (-4/5,3/5) ​

 

[petertheta]

There is another vector given but I assume this to be just another vector for which I must do the same transformation to and expressing as a linear combination of $$\hat{r} and \hat{\theta}$$ it's $$\vec{u} = 5\hat{x} + 0\hat{y}$$ so not the point PQ like you suggest.

P

 

[tiny-tim]

my guess is that the 5x + 0y vector is to start at P

 

[petertheta]

I've not seen this method before can you explicitly show me how to proceed?

In the question the vectors are the other way around so $$\vec{v1} = 5\hat{x} + 0\hat{y}$$ and $$\vec{v2} = 3\hat{x} + 4\hat{y}$$

Thanks

 

[tiny-tim]

good morning!

you transfer the origin from O to P,

then your second vector is expressed relative to the usual x,y axes,

and you need to express it relative to the two new axes along r and θ

 

[petertheta]

I'm afraid I've not covered this before so am at a loss of even how to start this?

 

[tiny-tim]

try a google search for "rotation of coordinates in 2D",

eg http://zone.ni.com/reference/en-XX/help/371361J-01/gmath/2d_cartesian_coordinate_rotation/