Givenanythreepointsa,b,cintheaffineplane A2, letting (a1,a2),(b1,b2),and(c1,c2) be the coordinates of a,b,c, with respect to the standard affine frame for A2, prove that a,b,c are collinear iff
i.e., the determin antis null. Letting (a0,a1,a2), (b0,b1,b2),and (c0,c1,c2) be the bary centric coordinates of a,b,c with respect to the standard affine frame for A2, prove that a,b,care collinear iff
Given any four points a,b,c,d in the affine space A3, letting (a1,a2,a3),(b1,b2,b3),(c1,c2,c3),and(d1,d2,d3) be the coordinates of a,b,c,d, with respect to the standard affine frame for A3, prove that a,b,c,d are coplanar iff
i.e., the determinant is null.
Letting(a0,a1,a2,a3),(b0,b1,b2,b3),(c0,c1,c2,c3),and(d0,d1,d2,d3)be the bary centric coordinates of a,b,c,d, with respect to the standard affine frame for A3,prove that a,b,c,d are coplanar iff