It is typical of projective geometry that quantities, such as planes, quadrics, or the like, are defined by the vanishing of some linear relation. Here, one five-dimensional coefficient vector represents the surface of a sphere, leaving a four-dimensional space of vectors which could represent points on the sphere. Since at most four of them are linearly independent, four points suffice to define a sphere.
Two spheres whose coefficients are not multiples of each other, define an intersection, which we know to be a circle. Three points must be found, from which to construct the circle. In generality, the problem is to partition space into two sets of linearly independent vectors; one of which holds given conditions, the other the points fulfilling them.
Projective geometry abounds with symbolism which describes the solutions, mostly in terms of cross products, determinants, cofactors, and similar entities. Numerically, it is simpler to start with a matrix of linearly independent vectors, and follow some procedure which will adapt the basis to the conditions at hand, without using all the vocabulary and symbolism of projective geometry.
As it happens, finding families of vectors whose inner products are mostly zero and occasionally not, results by inverting a matrix, which accounts for the presence of cofactors and their ilk in symbolic analyses. Thus, we need a procedure for completing and inverting partially defined matrices.
In the present situation, we have two conditions, namely the coefficients of the two spheres. We require three vectors, each orthogonal to the first two, namely the coordinates of three points which lie on the surfaces of both spheres; the remainder of the circle of intersection consists of linear combinations of these first three.