How Do You Convert Cartesian Vector Coordinates to Cylindrical Coordinates?

[EugP]

Homework Statement

Transform the following vector into cylindrical coordinates and then evaluate them at the indicated points:

\vec A = (x + y)\hat x

at

P_1 (1, 2, 3)

Homework Equations

r = \sqrt{x^2 + y^2}

\phi = \tan^{-1}(\frac{y}{x})

z = z

The Attempt at a Solution

r = \sqrt{x^2 + 0^2} = x

\phi = \tan^{-1}(\frac{0}{x}) = 0

z = z = 0

\vec A = x\hat r at point P_1 (1, 2, 3) \Longrightarrow \hat r

Could someone please check if this is correct? There are a few more of these, but if I can do this one, then the rest are no problem. Thanks.

 

[HallsofIvy]

Can I assume that \hat{x} is the unit vector in the x direction? If so then (x+ y)\hat{x} is not a "vector", it is a "vector field"- a vector at each point in the xy-plane. At (1, 2, 3) (surprising how often that point shows up!), that is the vector 3 \hat{x}, of length 3 pointing in the x-direction. That vector has no z-component. The projection of the vector <2, 0> in the direction of the <1, 2> vector will be the \hat{r} component. <2, 0> minus that projection will be the component in the \theta direction.