If Vector PO + OQ = Vector QO + OR, Prove That the Points P, Q, R Are Collinear

A standard vector algebra problem, not tied to a specific exam paper.

Problem

If $\overrightarrow{PO}+\overrightarrow{OQ}=\overrightarrow{QO}+\overrightarrow{OR}$, prove that the points $P$, $Q$, $R$ are collinear.

Note: the worked solution embedded below is titled $\overrightarrow{PQ}+\overrightarrow{OQ}=\overrightarrow{QO}+\overrightarrow{QR}$. That version does not force $P$, $Q$, $R$ to be collinear, so the standard form of the question, which the working itself follows, is used here.

Key insight. Writing every vector in the equation in terms of position vectors relative to the origin $O$ turns the given condition into a single equation relating the position vectors of $P$, $Q$, $R$ — and that equation says exactly that $Q$ is the midpoint of $P$ and $R$, which is more than enough to guarantee collinearity.

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Approach

Let $\vec p$, $\vec q$, $\vec r$ denote the position vectors of $P$, $Q$, $R$ relative to the origin $O$. Rewrite each of $\overrightarrow{PO}$, $\overrightarrow{OQ}$, $\overrightarrow{QO}$, $\overrightarrow{OR}$ in terms of $\vec p$, $\vec q$, $\vec r$, substitute into the given equation, and solve for $\vec q$ in terms of $\vec p$ and $\vec r$. If the result expresses $\vec q$ as a point that divides $PR$ in a fixed ratio, that’s exactly what it means for $P$, $Q$, $R$ to be collinear.

Solution

Step 1 — Write each vector in terms of position vectors

$$\overrightarrow{PO} = -\vec p, \qquad \overrightarrow{OQ} = \vec q, \qquad \overrightarrow{QO} = -\vec q, \qquad \overrightarrow{OR} = \vec r$$

Step 2 — Substitute into the given equation

$$\overrightarrow{PO}+\overrightarrow{OQ} = \overrightarrow{QO}+\overrightarrow{OR}$$

$$-\vec p+\vec q = -\vec q+\vec r$$

Step 3 — Solve for q

$$2\vec q = \vec p+\vec r$$

$$\vec q = \frac{\vec p+\vec r}2$$

Step 4 — Interpret this result

$\vec q=\dfrac{\vec p+\vec r}2$ is exactly the position vector of the midpoint of $P$ and $R$. Since the midpoint of two points always lies on the line joining them, $Q$ lies on line $PR$.

Step 5 — Conclude

Since $Q$ lies on the line through $P$ and $R$, the three points $P$, $Q$, $R$ are collinear.

Answer

Common mistakes

• Mishandling the signs of $\overrightarrow{PO}$ and $\overrightarrow{QO}$. Both are “reversed” vectors — $\overrightarrow{PO}=-\vec p$ and $\overrightarrow{QO}=-\vec q$ — since they point from $P$ (or $Q$) to the origin, not the other way round; swapping either sign changes the final relationship between $\vec p$, $\vec q$, $\vec r$ entirely.
• Stopping at $2\vec q=\vec p+\vec r$ without recognising what it means. The equation itself isn’t the proof — it has to be read as “$Q$ is the midpoint of $PR$,” which is what actually establishes collinearity (a midpoint always lies on its segment’s line).
• Confusing this with a general collinearity condition. Collinearity only requires $\vec q=\vec p+t(\vec r-\vec p)$ for some scalar $t$ — here the working happens to pin down the specific value $t=\dfrac12$, but the proof would be equally valid for any other value of $t$ that emerged from the algebra.

Practise next

• If $\overrightarrow{AO}+\overrightarrow{OB}=2\left(\overrightarrow{BO}+\overrightarrow{OC}\right)$, show that $B$ divides $AC$ in a fixed ratio and find that ratio.