# Transform Cylindrical coordinates into Cartesian Coordiantes

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1. Oct 8, 2014

### kexanie

I've learned that a vector in coordinate system can be expressed as follows:
A = axAx+ayAy+azAz.
ai, i = x, y, z, are the base vectors.
The transformation matrix from cylindrical coordinates to cartesian coordiantes is:
Ax cosΦ -sinΦ 0 Ar
Ay = sinΦ cosΦ 0 mutiplye by AΦ
Az 0 0 1 Az

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

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

2. Oct 8, 2014

### mathman

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

3. Oct 8, 2014

### kexanie

sorry, it should be . The formula was not inserted successfully.

Last edited: Oct 8, 2014
4. Oct 9, 2014

### 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.