I'd like to check to see if 3 points in $3D$ space are collinear, with a complication.
Instead of knowing the points, for each point I instead have a set 3 planes whose intersection defines the point. These planes are known to meet at a single point. So, to clarify, there are 9 planes total.
While it would be possible to find each intersection and then use the points to check for collinearity, is there an elegant way to check for it while still using the planes?
Naming the three points $A$, $B$, and $C$, my thought was to try and check if $|(B-A)\times(C-A)|$ was equal to zero, substituting in the solution for solving for the intersection of the planes using Cramer's rule and attempting to simplify from there. It's not looking very elegant however, so I was wondering if I'm missing something here.
In fact you problem belongs to algebraic geometry. In AG the only way to define a point is by defining it as the quotient of the ring in three variables by an ideal generated by three $1$st degree polynomials (your planes). The line connecting two points is then defined by the intersection of these ideals, in other words it is defined as the set of planes that belongs to the two ideals (i.e. the planes that pass through th two points). If $p_1, q_1, r_1$ are the polynomials (planes) defining the first point, and $p_2, q_2, r_2$ the polynomials defining the second point then the planes throug the connecting line are determined by the coefficients $\alpha, \beta, \ldots$ such that $\alpha p_1 + \beta q_1 + \gamma r_1 = \delta p_2 + \eta q_2 + \theta r_2$. If the intersection of the set of these planes with the ideal corresponding to the third point is not a unique plane then the three points are colinear. I added the tag algebraic geometry.