Cylindrical Coordinates

Hello,

I am reading Electromagnetic Fields, 2nd Edition by Ronald K. Wangsness, and doing its exercises.

I have problem solving Question 1-19, as following:
Question:
Given vector A= a*ρ+b*φ+c*z, where ρ, φ and z are the unit vectors of cylindrical coordinates. Find the rectangular components of A, expressing them in terms of x, y and z, respectively.

The answer given at the end of the book is:
(x^2+y^2)^(-1/2)*[(ax-by)*i+(ay+bx)*j]+c*k, where i, j and k are the unit vectors of rectangular coordinates.

I really dont know how to get this answer... somebody please help!


And more over, I am rather confused by the whole concept of cylindrical coordinates. For e.g., if, in cylindrical coordinates, a point P=P(2, pi/4,1), then the vector starting from O to P, i.e. OP should be
vector OP=2*ρ+pi/4*φ+1*z
or
vector OP=2*ρ+1*z
?
Thanks a lot!

hi

Given a cartesian vector
A = Ax ax + Ay ay + Az az

We need a vector in cylindrical coordinates
A = Aρ aρ + Aφ aφ + Az az

To find any desired component of a vector
Aρ = A . aρ and Aφ = A . aφ

Expanding these dot products, we have
Aρ = (Ax ax + Ay ay + Az az) . aρ = Ax ax.aρ+Ayay.aρ

Aφ = (Ax ax + Ay ay + Az az) . aφ = Ax ax.aφ+Ayay.aφ

and

Az = (Ax ax + Ay ay + Az az) . az = Az az.az = Az

So in order to complete the transformation, it is necessary to know the dot products ax.aρ, ay.aρ, ax.aφ and ay.aφ.. the result will be the angle between the two unit vectors in question which is easy to make and also you can find a table of conversion in your text book

thank you very much! it helped !