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Prove some properties of a plane determined by points P,Q,R

Let P,Q,R be three points in \mathbb{R}^3 not all on the same line, and let M be the plane they determine.

  1. If p,q,r \in \mathbb{R} with p+q+r = 1 prove that the point pP + qQ + rR is on M.
  2. Prove that all of the points on M are of the form pP + qQ + rR for p,q,r \in \mathbb{R} with p+q+r = 1.

  1. Proof. Since M is the plane determined by the three points P,Q,R we know M is the set of points

        \[ M = \{ P + s(Q-P) + t(R-P) \} \qquad s,t \in \mathbb{R}. \]

    Let s = q, \ t = r. Then,

        \begin{align*}  P + q(Q-P) + r(R-P) \in M && \implies && P - qP - rR + qQ + rR &\in M \\  && \implies && (1-q-r)P + qQ + rR &\in M \\  && \implies && pP + qQ + rR &\in M \end{align*}

    since p+q+r = 1 implies p= 1 - q - r. \qquad \blacksquare

  2. Proof. Let X be a point on M. Then, we know there exist s,t \in \mathbb{R} such that

        \begin{align*}  && X &= P + s(Q-P) + t(R-P) \\  \implies && X &= (1-s-t)P + sQ + tR. \end{align*}

    But, 1-s-t + s+t = 1; thus, X = pP + qQ + rR for p = 1-s-t, \ q = s, \ r = t. \qquad \blacksquare

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