Goemans and Williamson simply choose a uniformly random hyperplane through the origin and divide the vertices according to which side of the hyperplane the corresponding vectors lie.

Let , and be the corresponding vectors and matrix.

At the level of root systems the direction corresponds to pointing towards the shorter vector; edges labeled "3" have no direction because the corresponding vectors must have equal length.

The three corresponding normal vectors of the corner's sides form a basis (x, y, z) in which to represent the direction of an arbitrary incoming ray, [a, b, c].

The dihedral angle between two non-parallel hyperplanes of a Euclidean space is the angle between the corresponding normal vectors.

Entries in the interior give the product of the corresponding vectors on the left and the top in that order (the product is anti-commutative).

It also means that if there are several vanishing singular values, any linear combination of the corresponding right-singular vectors is a valid solution.

We re required to find the eigenvalues i which satisfy, and the corresponding vectors.

The -superscripted symbols denote the cartesian components of the corresponding vectors.

An important step in preparation for calculating in perturbative quantum field theory is to separate the "operator" factors and above from their corresponding vector or spinor factors and .