Transform Cylindrical coordinates into Cartesian Coordiantes

[kexanie]

I've learned that a vector in coordinate system can be expressed as follows:
A = a_xA_x+a_yA_y+a_zA_z.
a_i, i = x, y, z, are the base vectors.
The transformation matrix from cylindrical coordinates to cartesian coordiantes is:
A_x cosΦ -sinΦ 0 A_r
A_y = sinΦ cosΦ 0 mutiplye by A_Φ
A_z 0 0 1 A_z

and the conversion formula
x = rcosΦ
y = rsinΦ
z = z

1. What's the difference between this two kind of equations?
2. Why A_x is not equal to x?
3. I was told that A_x might be a function of x, y and z. Is the latter kind of equaltions has a prerequisite that a_x = (1, 0, 0), but in the first kind of equations, the base vector can be anything else?
4. From the matrix, A_x = cosΦA_r - sinΦA_Φ, that is not equal to x = rcosΦ !? Why? How should I apply the transformation matrix?

Thanks in advance.

 

[mathman]

Ax cosΦ -sinΦ 0 Ar
Ay = sinΦ cosΦ 0 mutiplye by AΦ
Az 0 0 1 Az
Above is confusing - looks like typos.

 

[kexanie]

sorry, it should be

. The formula was not inserted successfully.

 

[mathman]

The matrix formulation looks like a rotation of the (x,y) coordinates around the z axis through an angle φ, not a conversion from cylindrical to cartesian coordinates.

 

[blue_raver22]

I can provide a response to your questions.

The difference between the two equations is that the first one is a vector representation in cylindrical coordinates, while the second one is a conversion formula from cylindrical to Cartesian coordinates. In the first equation, Ax is the component of the vector A in the x-direction, while in the second equation, x represents the x-coordinate in the Cartesian system.

Ax is not equal to x because in the first equation, Ax is a vector component, while in the second equation, x is a coordinate. They have different meanings and cannot be equated.

The base vectors in the first equation (ax, ay, az) can be any set of mutually perpendicular unit vectors, as long as they are consistent with the chosen coordinate system. In the second equation, the base vectors are the unit vectors in the x, y, and z directions.

The matrix transformation is used to convert the components of a vector from cylindrical coordinates to Cartesian coordinates. It is not meant to be applied to the conversion formula. The conversion formula should be used to convert the coordinates of a point from cylindrical to Cartesian.

I hope this helps clarify any confusion. If you have any further questions, please feel free to ask.