Conversion vectors in cylindrical to cartesian coordinates

[ForTheGreater]

Homework Statement

It's just an example in the textbook. A vector in cylindrical coordinates.
A=a_rA_r+a_ΦA_Φ+a_zA_z
to be expressed in cartesian coordinates.
Start with the A_x component:
A_x=A⋅a_x=A_ra_r⋅a_x+A_Φa_Φ⋅a_x

a_r⋅a_x=cosΦ
a_Φ⋅a_x=-sinΦ

A_x=A_rcosΦ - A_ΦsinΦ

Looking at a figure of the unit vectors I get it. At the same time I just don't understand why A_rcosΦ isn't enough to get the magnitude of the A_x component.

 

[kuruman]

Because unit vector ##\hat{x}## has two components, one in the ##\hat{r}## direction and one in the ##\hat{\phi}## direction both of which depend on the angle ##\phi##. So when you express ##\hat{x}## as in ##A_x\hat{x}##, in terms of cylindrical unit vectors, you get two components which means you need the Pythagorean theorem to find the magnitude of ##A_x##.

 

[ForTheGreater]

If I have a point r=4, phi=pi/8 and z=z; in cylindrical coordinates.
sqrt( (r*cos(phi) )² + (r*sin(phi) )² ) = 4
So taking r*cos(phi) as the x component and r*sin(phi) as the y component seems to be enough to get the same point represented in both coordinate systems? What am I missing?

 

[kuruman]

Suppose I gave you vector
$$\vec{A}=3\hat{a}_r-2 \hat{a}_{\phi}+4 \hat{z}$$
How would you proceed to find the Cartesian x-component of this vector? You will need the equations that transform the cylindrical unit vectors into the Cartesian unit vectors.

 

[Chestermiller]

Homework Statement

It's just an example in the textbook. A vector in cylindrical coordinates.
A=a_rA_r+a_ΦA_Φ+a_zA_z
to be expressed in cartesian coordinates.
Start with the A_x component:
A_x=A⋅a_x=A_ra_r⋅a_x+A_Φa_Φ⋅a_x

a_r⋅a_x=cosΦ
a_Φ⋅a_x=-sinΦ

A_x=A_rcosΦ - A_ΦsinΦ

Looking at a figure of the unit vectors I get it. At the same time I just don't understand why A_rcosΦ isn't enough to get the magnitude of the A_x component.

You did this correctly. In terms of interpretation, ##A_r \cos {\phi}## is only the component of the r component of A in the x direction. You also need the component of the ##\phi## component of A in the x direction. This is your ##-A_{\phi}s\sin{\phi}##

 

[ForTheGreater]

Thank you, I think I just didn't think of that you'll need both the Ax component and the Ay component of the vector to get the x coordinate.

 

[Chestermiller]

Thank you, I think I just didn't think of that you'll need both the Ax component and the Ay component of the vector to get the x coordinate.

I think you meant Ar and Atheta