A physical quantity can be represented by the following form:
$A = a_1\sigma_1 + a_2\sigma_2 + a_3\sigma_3$ where $\sigma$ matrices are Pauli matrices. Also suppose that there is $B = b_1\Sigma_1 + b_2\Sigma_2 + b_3\Sigma_3$.
A. I read that $a_1,a_2,a_3$ are direction vectors, and when the sum of A's representing real quantity and B's is zero, as each quantity can only be 1 or -1, probability $AB$ being 1 or -1 depend on the value of $-a_1b_1 - a_2b_2 - a_3b_3$.
The question would be: isn't $a_1,a_2.., b_1,b_2,..$ only direction vectors? If so, how can probability of finding $AB$ being 1 change? I do get this mathematically, but I am unable to picture the situation with $a,b$ being direction vectors.
B. I read that when the sum of A's representing real quantity and B's is zero, determining a value on one direction of $A$ set the value of the whole $A$. Is this true? Can anyone explain this?