Prove that the line through two distinct points on a plane is entirely contained in the plane

by RoRi

April 29, 2016

Let $M$ be a plane containing two distinct points $P$ and $Q$ . Prove that every point on the line through $P$ and $Q$ is in $M$ .

Proof. Fist, the line through $P$ and $Q$ is given by

$L = \{ P + r(Q-P) \}.$

Then, let $R$ be any point on $M$ not on $L(P;Q)$ . The plane $M$ is given by

$M = \{ P + s(Q-P) + t(R-P) \}.$

So, for any point $X \in L(P;Q)$ let $s = r$ and $t = 0$ . Then we have

$P + s(Q-P) + t(R-P) = P + r(Q-P) \in M \quad \implies \quad L(P;Q) \in M. \qquad \blacksquare$

Stumbling Robot