I don't understand this problem. This is supposedly a review question from calc 3, but I've never seen anything like this before. Fortunately, I solved it, but I just don't understand what I did. (It's probably the weird notation I'm not used to.)

Express the r-component, $\displaystyle A_r$, or a vectorAat $\displaystyle ($$\displaystyle r_1, $$\displaystyle \phi_1$$\displaystyle , z_1$$\displaystyle )$

(A) In terms of $\displaystyle A_x$ and $\displaystyle A_y$ in Cartesian coordinates, and

(B) in terms of $\displaystyle A_r$ and $\displaystyle A_\theta$ in spherical coordinates.

My book says:

"The vector $\displaystyle A$ in Cartesian coordinates with components $\displaystyle A_x, A_y, and A_z$ can be written as:

$\displaystyle A$$\displaystyle = a_x$$\displaystyle A_x +$$\displaystyle a_y$$\displaystyle A_y +$$\displaystyle a_z$$\displaystyle A_z$

Answers:

(A) $\displaystyle A_x cos(\phi_1) + A_y sin(\phi_1)$

(B) $\displaystyle A_r \frac{r_1}{\sqrt(r_1^2+z_1^2)} + A_\theta \frac{r_1}{\sqrt(r_1^2+z_1^2)}$

So for part (A), is that written in the same form as the unit vectors i hat, j hat, and k hat? And for part (B), why are both components divided by the radius?