converting force from spherical to Cartesian coords

Question

So I am working on my assignment, and have a question about converting coordinates. I dont know whether I should ask here or the math SE, so lets give it a try here.

The force in question is $$\vec{F} = -k\,r^{-n}\,\hat{r}$$

I know the conversion equations but I have no idea how to do it. I need to split the $\hat{r}$ into its corresponding i, j, k vectors. I wanna take the curl to make sure that its zero and hence the force is conservative.

Note: the question asks to be converted. I know how to already do it in spherical coordinates.

Answer 1

$$r^2 = x^2 + y^2 + z^2$$

What does that make $r^{-n}$?

As for converting $\hat{r}$, the position vector can be written

$$\vec{r} = r \hat{r}$$

in spherical coordinates, but it can also be written

$$\vec{r} = x\hat{i} + y\hat{j} + z\hat{k}$$

in rectangular coordinates. Therefore, the two are equal.

$$r \hat{r} = x\hat{i} + y\hat{j} + z\hat{k}$$

From there you can finish up finding a formula for $\hat{r}$ in rectangular coordinates.