Grassmann Algebra
Hermann Grassmann had the notion that it was possible to use algebra to prove geometric facts. Instead of using scalar Cartesian coordinates, he used actual points in space. Let E be Euclidean space and call the elements of E points. If P and Q are two points then PQ should represent something like the line from P to Q. If P, Q, and R are three points then PQR should resemble the triangle having vertices P, Q, and R. And so on.
If t is a scalar tP is the point P with weight t. Let R(t) = (1 – t)P + tQ, where P and Q are points and t is a scalar, then R(0) = P and R(1) = Q. In general aP + bQ is the point (a/(a + b) P + b/(a + b) Q = R(b/(a + b)) having weight a + b.
What is PQ if P = Q? If PQ resembles the line segment from P to Q, then the line from P to P is the point P. Assuming P = P2 we have P + Q = (P + Q)(P + Q) = PP + PQ + QP + QQ = P + PQ + QP + Q so 0 = PQ + QP and hence 0 = PP + PP = 2P for any point P. That might have been the first theory Hermann discarded before he came up with the idea PQ = 0 if P = Q. We have 0 = (P + Q)(P + Q) = PQ + QP so PQ = -QP. Since 0 = -0 we don’t have to discard this theory yet.
Note that PQR = 0 if P = Q or Q = R or R = P. More generally PQR(t) = 0 for all t with R(t) as above. In general, PQR = 0 if and only if the three points are colinear.
For example, P(Q + R) is the median of the triangle PQR from P to the midpoint of QR. The barycenter of the triangle is P + Q + R. Note P(Q + R)(P + Q + R) = PQP + PQQ + PQR + PRP + PRQ + PRR = PQR + PRQ = 0. This proves the medians of any triangle meet at a point, if you think about the symmetry for a second.
Grassman’s rule that P2 = 0 enables the detection of points not being in general position, i.e., they don’t determine an n – 1 dimensional object. If n points belong to a subspace of dimension less than n – 1 their product vanishes, not just for n = 2