What is the Meaning of Cylindrical Vectors in Relation to Cartesian Vectors?

[salman213]

1. IM very confused about the meaning of these cylindrical vectors

for Cartesian vectors if i say A = 1ax + 2ay + 3az

I know i mean a vector with a magntiude of 1 in the x direction 2 in the y direction and 3 in the z direction and i make a line from the origin to point (1,2,3).

Now for cylindrical I cannot think the same way

for a point i can make a point easily it seems to have a similar meaning (r is the magntiude from origin to the point, phi being the angle from x and z being the regular z)


now for a vector example a = 1 ap + 2 aphi + 3 az


Im very confused how do i draw this vector and what does it mean?



the magntiude along p is 1 so the angle is 2? from the x axis?

i don't think this is correct.. I can't understand this help!

 

[chrisk]

Cylindrical coordinates can be thought of as a right triangle with a point of rotation at the origin. The one leg of the triangle lies in the x-y plane and is

$$\rho$$

and the vertical leg is

$$\mbox{z}$$

and the angle with respect to an axis, usually the x axis, that the triangle is rotated from is

$$\phi$$

These three values form another orthogonal coordinate system but it is not fixed like a Cartesian coordinate system but changes direction as the point changes position.

 

[salman213]

so you would not be able to draw the vector out?

how would i draw or specifiy a change in aphi like on a 3d plane what would the difference of

1 ap + 2aphi + 3az and 1 ap + 3aphi + 3az ?

 

[chrisk]

View the first image of this link to see how a cylindrical coordinate system relates to a Cartesian coordinate sytem. I corrected my earlier post so read it again.

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

 

[salman213]

the thing is this all seems to be about points i don't understand how to relate it to vectors in cylindrical system

I was wondering if someone can give me an idea of how to represent a vector in cylindrical coordinate system.

like on a 3d plane what would the difference of

1 ap + 2aphi + 3az and 1 ap + 3aphi + 3az ?

 

[chrisk]

I think what you are looking for is a relation between the Cartesian vectors and the Cylindrical vectors. So, using the picture from the link given previously and some trigonometry, we have

$$\vec{x}=\rho\mbox{cos\phi}\hat{x}$$

$$\vec{y}=\rho\mbox{sin\phi}\hat{y}$$

$$\vec{z}=z\hat{z}$$

where

$$\hat{x},\hat{y}, and \ \hat{z}$$

are unit vectors in the x, y, and z directions. The inverse relations are

$$\vec{\rho}=\sqrt{x^2+y^2}\hat{\rho}$$

$$\vec{\phi}=\arctan{\frac{y}{x}}\hat{\phi}$$

The z value is the same for both coordinate systems. Hope this helps.