Coplanarity

Remarks

When $A$ has dimension at least $2$, the set $S$ is coplanar exactly when some affine plane of $A$ contains $S$.

Example: in $\R^3$ the points $(0,0,0)$, $(1,0,0)$, $(0,1,0)$ and $(1,1,0)$ are coplanar. Check: all differences lie in the subspace $\set{z = 0}$, which has dimension $2$.

Non-example: the points $(0,0,0)$, $(1,0,0)$, $(0,1,0)$ and $(0,0,1)$ of $\R^3$ are not coplanar. The differences from the origin are the three standard basis vectors, which span a space of dimension $3$.

In $\R^3$, four points $p_0, p_1, p_2, p_3$ are coplanar exactly when the determinant of the $3 \times 3$ matrix with columns $p_1 - p_0$, $p_2 - p_0$, $p_3 - p_0$ is zero. Every set of at most three points is coplanar.