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.