The eigenvectors of the first Pauli matrix encode
— as it were — the +X and −X directions. Likewise, the eigenvectors of the second Pauli matrix encode
the +Y and −Y directions and those of the third Pauli matrix, the +Z and −Z directions. The dot product of the Pauli vector with any unit vector ⃑N yields a matrix which likewise has eigenvalues +1 and −1 and a pair of eigenvectors; the eigenvector with positive eigenvalue encodes
the +N direction and the eigenvector with negative eigenvalue encodes
the −N direction. The dot product of any two such encoded
directions yields the probability amplitude that the spin of an electron prepared in the first direction will collapse to the second direction when measured along that second direction.