Context: We were taught collinearity of 3 points (in vectors) and a method for checking 4 points for coplanarity simultaneously. Searching around I found a better method for checking for coplanarity in 4 points i.e. the 4x4 determinant.
In a 2d plane there is a 3x3 determinant to check for collinearity of 3 points In a 3d plane similarly a 4x4 determinant for coplanarity of 4 points So for 5 points would it have to be a 4d plane? If so how does that work?
There is already a simple way that I thought of to check for coplanarity in any number points which would be to check $\binom{n}4$ cases of coplanarity many of which are actually not needed for eg: in 5 points you would only need to check 2 cases out of 5 from the 4x4 det and 3 cases out of 10 from the 3x3 det.
So is there a way to check directly (without forming cases) the coplanarity of 5 points in a 3d coordinate system (maybe like a 5x5 determinant?)