Why are Pauli vectors for different particles opposite in the singlet state? If $\boldsymbol{\sigma_{1}}$ is the Pauli vector for a particle and $\boldsymbol{\sigma_{2}}$ for the other particle, why is $\boldsymbol{\sigma_{1}}=-\boldsymbol{\sigma_{2}}$ in the singlet state? I read this in a book and I think there's something fundamental that I'm missing.
What would happen in the triplet states?
Edit:
I'm trying to prove that the operator $S_{12}=3(\boldsymbol{\sigma_{1}}\cdot\boldsymbol{\hat{r}})(\boldsymbol{\sigma_{2}}\cdot\boldsymbol{\hat{r}})-\boldsymbol{\sigma_{1}}\cdot\boldsymbol{\sigma_{2}}$ is $0$ acting on a singlet state (and trying to calculate its eigenvalues in the triplet state). If $\boldsymbol{\sigma_{1}}=-\boldsymbol{\sigma_{2}}$ then $S_{12}=0$.
Page 77 of this book (Bertulani, Carlos A. Nuclear Physics in a Nutshell. Princeton University Press, 2007.): https://books.google.es/books?id=n51yJr4b_oQC&printsec=frontcover#v=onepage&q&f=false
 A: The eigenvalues of unit Pauli vectors are $\pm 1$ .
The easiest way for you to appreciate the structure is by inspection of the singlet and triplet states in the z-direction,  whose highest - $\sigma_z$ states are
$$
 s = (\uparrow \downarrow -  \downarrow\uparrow   )/\sqrt{2}, \qquad t= \uparrow  \uparrow  .
$$
(I somewhat perversely chose different $\sigma_z$ eigenstates to remind you the second piece of operator S is rotationally invariant, and for ease of computation below!)
By rotational invariance, you may choose 
$$\boldsymbol{\hat{r}}= \boldsymbol{\hat{z}},\Longrightarrow (\boldsymbol{\sigma_{1}}\cdot\boldsymbol{\hat{r}})(\boldsymbol{\sigma_{2}}\cdot\boldsymbol{\hat{r}})=\sigma_1^z \sigma_2^z,$$
 with eigenvalues -1 for the singlet and 1 for the triplet.
The other term (Dirac exchange operator) resolves to the usual ladder operator (work them out!)  and z-components of quadratic Casimir invariants,
$$
 \boldsymbol{\sigma_{1}}\cdot\boldsymbol{\sigma_{2}}= \sigma_1^z \sigma_2^z + (\sigma_1^+ \sigma_2^-  + \sigma_1^- \sigma_2^+)/2 ,$$
with evident eigenvalues -3 for the singlet and 1 for the triplet.
Your operator  S then  has null eigenvalue  for the singlet and eigenvalue 2 for the triplet.
