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.

 

[Architeuthis Dux]

Your solution is correct! Just make sure to include the units for each coordinate (e.g. r is in meters, \phi is in radians, and z is in meters). Also, don't forget to convert the vector components to the appropriate units if necessary. Great job!