vector - Determine coplanar (or collinear) points in higher dimensions with Java -

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in java, best way determine if point (of have many) given in 6 dimensions lies on same plane other points in 6-dimensional space? it's clear how in 2d , 3d, far know of concepts such cross product, etc, not generalise higher dimensions (or specific few).

it helpful find way answer same question points on lines in 6d. more general, have given higher dimensional point cloud , want determine if of these points lie on same plane.

in n-dimensional space, point lies in same hyperplane n (non-degenerate) others, if volume of simplex, formed these (n+1) points, zero. simplex volume calculated through determinant

          |1 x1 y1 z1 ...|           |1 x2 y2 z2 ...| v = 1/n! |1 x3 y3 z3 ...|           |1 x4 y4 z4 ...|           |..............|

note formula corresponds cross product check collinearity in 2d case, mixed product check coplanarity in 3d case etc


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