The Physical System (Background)
The system in Box 2.7 is a spin singlet, which physically refers to two electrons coupled such that their spin angular momentum cancels out perfectly. In many regards, two electrons in this state behave as a single particle with zero spin angular momentum.
The angular momentum of a classical particle has a continuous spectrum of possible directions and magnitude. In stark contrast, electron spin has only two possible directions (up and down) and one possible magnitude ($\frac{1}{2} \hbar$). In a spin singlet state one electron has up-spin, and the other has down-spin.
When two electrons in a singlet state are spatially separated, they continue to behave as a single particle in certain ways. This is the type of the non-local behavior that made many physicists uneasy (notably the EPR authors).
The combination of spin and electric charge in an electron gives it a magnetic dipole moment. To determine the spin state of an electron, Stern-Gerlach measurements pass electrons through an inhomogeneous magnetic field such that electrons experience a force dependent on the orientation of their magnetic moment. The result is that electrons with up and down spin are deflected in two distinct directions.
Choice of Measurement Basis, $\vec v \cdot \vec \sigma$ (Answer)
When Alice and Bob perform a Stern-Gerlach measurement on their respective electrons, they must each choose an orientation of the magnetic field described above. In other words they must pick a $\vec v$, which is equivalent to picking a point on a unit sphere. Here, the Pauli matrices ($\vec \sigma$) can be thought of as a convenient basis for real 3-space.
In your example, Alice has chosen
$$A = \vec v \cdot \vec \sigma = \frac{1}{\sqrt{38}} (2,3,5) \cdot \vec \sigma = \frac{1}{\sqrt{38}} \begin{bmatrix} 5 & 2-3i \\ 2+3i & -5 \end{bmatrix}.$$
This matrix is called a single particle spin operator and represents the orientation of Alice's magnetic field relative to some agreed coordinate reference frame. This matrix is Hermitian, which means that it is "real" (as opposed to complex or imaginary) in every way that matters. Bob's decision to choose the same orientation means that the joint spin measurement operator is given by
$$A \otimes A = \frac{1}{38}\begin{bmatrix} 25 & 10-15i & 10-15i & -5-12i \\ 10+15i & -25 & 13 & -10+15i \\ 10+15i & 13 & -25 &-10+15i \\ -5+12i & -10-15i & -10-15i & 25 \end{bmatrix}.$$
The wavefunction of a singlet states is given by
$$\vert \psi \rangle = \frac{\vert 01 \rangle- \vert 10 \rangle}{\sqrt 2}= \begin{bmatrix} 0 \\ \frac{1}{\sqrt{2}} \\ -\frac{1}{\sqrt{2}} \\ 0 \end{bmatrix}.$$
Now we have everything we need to evaluate the wavefunction over the joint spin measurement operator with nothing more than matrix multiplication.
$$\langle \psi \vert A \otimes A \vert \psi \rangle = -1,$$
where the value $-1$ represents perfect anti-correlation (as claimed in Box 2.7).
It's worth noting that this result, and any others obtained in the same manner, is not sensitive to the coordinate reference frame agreed upon by Alice and Bob. As constructed any (non-relativistic) change in coordinate reference frame is effected by a unitary transformation, $A' = U^\dagger A U$, and unitary transformations preserve inner products.
If you want to go through the calculations, here's the Matlab script I used to double check it myself.
>> A=1/38^.5*[5,2-3i;2+3i,-5]
A =
0.8111 + 0.0000i 0.3244 - 0.4867i
0.3244 + 0.4867i -0.8111 + 0.0000i
>> AoA=kron(A,A)
AoA =
0.6579 + 0.0000i 0.2632 - 0.3947i 0.2632 - 0.3947i -0.1316 - 0.3158i
0.2632 + 0.3947i -0.6579 + 0.0000i 0.3421 + 0.0000i -0.2632 + 0.3947i
0.2632 + 0.3947i 0.3421 + 0.0000i -0.6579 + 0.0000i -0.2632 + 0.3947i
-0.1316 + 0.3158i -0.2632 - 0.3947i -0.2632 - 0.3947i 0.6579 + 0.0000i
>> psi=1/2^.5*[0;1;-1;0]
psi =
0
0.7071
-0.7071
0
>> psi'*AoA*psi
ans =
-1.0000
