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 What's the difference between this two kind of equations? Why Ax is not equal to x? 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? From the matrix, Ax = cosΦAr - sinΦAΦ, that is not equal to x = rcosΦ !? Why? How should I apply the transformation matrix? Thanks in advance.