Vector question: dot product in cylindrical coordinates

[JolileChat]

Hello.

To find the work of a force, I have to perform a dot product between the force and a infinitesimal displacement. If they are in cylindrical coordinates, I can't manage to make the dot product.

Please, could you help me?

Thank you.

[berkeman]

I googled dot product in cylindrical coordinates, and got lots of good hits. Here's one (the dot product in cylindrical coordinates is mid-way down the page):

http://www.iancgbell.clara.net/maths/vectors.htm

[JolileChat]

But the author of this page seems to have used a conversion to rectangular coordinates to get the result.

I learned that dot product is the sum of the product of the components of the two vectors. But if you multiply the angles, what should it give? The sum of the product of the components does not hold for cylindrical coordinates?