How to Construct the Spin Operator in an Arbitrary Direction?

[Matterwave]

Homework Statement

Construct the $$S_r$$ operator, which is the spin operator in the r direction.

Homework Equations

The pauli-matrices are useful, I would assume.

The Attempt at a Solution

So, the way we constructed the other 3 operators by noting their effects on a state |s m>

But I can't do that here because I have no idea what this operator $$S_r$$ will do to a state |s m>

I thought about looking at the fact that:

$$x = rsin(\theta)cos(\phi), y = rsin(\theta)cos(\phi), and z=rcos(\theta)$$

and just replace all the x's with $$S_x$$'s and r's with $$S_r$$'s etc. But I don't think this will work as it will give me 3 different answers.

I need help on how to start with this.

 

[Thaakisfox]

If you have a vector A then how do you calculate its component in a certain direction n?

You take the inner product of the unitvector of the given direction n and the vector A.

So this is the same case. How would you do it here? ;)