# Use of vectors to prove geometrical properties

> Edexcel International GCSE Further Pure Mathematics · 4PM1
> Source: https://www.owlsprep.com/study/edexcel-igcse-further-pure-math-s7-use-of-vectors-to-prove/

This guide teaches you to apply vector rules to prove core geometric properties for your Edexcel IGCSE Further Pure Math (4PM1) exam, including parallel lines, collinearity, concurrency, and ratio division of line segments.

**Prerequisites:** [Understanding of position vectors and vector arithmetic (addition, subtraction, scalar multiplication)](https://www.owlsprep.com/study/edexcel-igcse-further-pure-math-s7-position-vectors-and-displacement/)

## Learning objectives

- Prove two lines are parallel using vector scalar multiple relations
- Prove three points are collinear using shared point and parallel vector arguments
- Calculate the position vector of a point dividing a line segment in ratio m:n using the section formula
- Prove three lines are concurrent by showing they share a common intersection point

## Proving Parallel Lines Using Vectors

**Parallel Vectors** — Two vectors are parallel if one is a non-zero scalar multiple of the other. If $\overrightarrow{AB} = \lambda \overrightarrow{CD}$ for some non-zero constant $\lambda \in \mathbb{R}$, then line AB is parallel to line CD.

**Worked example:** Points A, B, C, D have position vectors $\mathbf{a} = \begin{pmatrix}1 \\ 3\end{pmatrix}$, $\mathbf{b} = \begin{pmatrix}5 \\ 7\end{pmatrix}$, $\mathbf{c} = \begin{pmatrix}2 \\ -2\end{pmatrix}$, $\mathbf{d} = \begin{pmatrix}8 \\ 4\end{pmatrix}$. Prove line AB is parallel to line CD.

1. Calculate vector $\overrightarrow{AB}$ using the position vector rule:

   $$\overrightarrow{AB} = \mathbf{b} - \mathbf{a} = \begin{pmatrix}5-1 \\7-3\end{pmatrix} = \begin{pmatrix}4 \\4\end{pmatrix}$$
2. Calculate vector $\overrightarrow{CD}$:

   $$\overrightarrow{CD} = \mathbf{d} - \mathbf{c} = \begin{pmatrix}8-2 \\4 - (-2)\end{pmatrix} = \begin{pmatrix}6 \\6\end{pmatrix}$$
3. Identify the scalar multiple relation between the two vectors:

   $$\overrightarrow{CD} = 1.5 \times \overrightarrow{AB}$$
4. Since $\overrightarrow{CD}$ is a non-zero scalar multiple of $\overrightarrow{AB}$, lines AB and CD are parallel.

> **Exam tip:** Always explicitly write the scalar multiple value in your proof, do not skip this step as it is required for full marks.

*Calculator:* allowed

## Proving Collinearity of Three Points

**Collinear Points** — Three or more points lie on the same straight line (collinear) if two vectors between pairs of the points share a common point and are parallel.

**Worked example:** Points P, Q, R have position vectors $\mathbf{p} = \begin{pmatrix}2 \\1\end{pmatrix}$, $\mathbf{q} = \begin{pmatrix}6 \\3\end{pmatrix}$, $\mathbf{r} = \begin{pmatrix}12 \\6\end{pmatrix}$. Prove P, Q, R are collinear.

1. Calculate vector $\overrightarrow{PQ}$:

   $$\overrightarrow{PQ} = \mathbf{q} - \mathbf{p} = \begin{pmatrix}6-2 \\3-1\end{pmatrix} = \begin{pmatrix}4 \\2\end{pmatrix}$$
2. Calculate vector $\overrightarrow{QR}$:

   $$\overrightarrow{QR} = \mathbf{r} - \mathbf{q} = \begin{pmatrix}12-6 \\6-3\end{pmatrix} = \begin{pmatrix}6 \\3\end{pmatrix}$$
3. Confirm parallelism:

   $$\overrightarrow{QR} = 1.5 \times \overrightarrow{PQ}$$
4. State the common point: vectors $\overrightarrow{PQ}$ and $\overrightarrow{QR}$ share the point Q, so P, Q, R lie on the same straight line and are collinear.

> **Exam tip:** Never forget to mention the shared common point in your collinearity proof; proving parallelism alone will only get you half marks.

*Calculator:* allowed

## Position Vector of a Point Dividing a Line in Ratio m:n

**Section Formula (Internal Ratio Division)** — If point M divides the line segment joining A (position vector $\mathbf{a}$) and B (position vector $\mathbf{b}$) in the ratio $m:n$ ($AM:MB = m:n$), then the position vector of M is $\mathbf{m} = \frac{n\mathbf{a} + m\mathbf{b}}{m + n}$.

**Worked example:** Point K divides the line joining A(1, 5) and B(7, 2) in the ratio 2:1. Find the position vector of K.

1. Identify values for the formula: $m=2$, $n=1$, $\mathbf{a} = \begin{pmatrix}1 \\5\end{pmatrix}$, $\mathbf{b} = \begin{pmatrix}7 \\2\end{pmatrix}$
2. Substitute into the section formula:

   $$\mathbf{k} = \frac{1\times\begin{pmatrix}1 \\5\end{pmatrix} + 2\times\begin{pmatrix}7 \\2\end{pmatrix}}{2+1} = \frac{\begin{pmatrix}1 + 14 \\5 +4\end{pmatrix}}{3}$$
3. Simplify to get the final position vector:

   $$\mathbf{k} = \begin{pmatrix}5 \\3\end{pmatrix}$$

> **Exam tip:** The section formula is *not* provided on the 4PM1 formula sheet, so you must memorize it. Note that the ratio values flip: the multiplier for $\mathbf{a}$ is the second value in the ratio, and the multiplier for $\mathbf{b}$ is the first value.

*Calculator:* allowed

## Proving Concurrency of Lines

**Concurrent Lines** — Three or more lines are concurrent if they all pass through a single common intersection point.

**Worked example:** In triangle ABC, points D, E, F are midpoints of AB, BC, CA respectively. Prove the lines AE, BF, CD are concurrent (meet at the centroid).

1. Let position vectors of A, B, C be $\mathbf{a}$, $\mathbf{b}$, $\mathbf{c}$ respectively. Write position vectors for midpoints:
2. $$\mathbf{e} = \frac{\mathbf{b} + \mathbf{c}}{2}, \mathbf{f} = \frac{\mathbf{a} + \mathbf{c}}{2}, \mathbf{d} = \frac{\mathbf{a} + \mathbf{b}}{2}$$
3. Write the parametric equation of line AE (scalar parameter t):

   $$\mathbf{r} = \mathbf{a} + t\left(\frac{\mathbf{b} + \mathbf{c}}{2} - \mathbf{a}\right) = (1-t)\mathbf{a} + \frac{t}{2}\mathbf{b} + \frac{t}{2}\mathbf{c}$$
4. Write the parametric equation of line BF (scalar parameter s):

   $$\mathbf{r} = \mathbf{b} + s\left(\frac{\mathbf{a} + \mathbf{c}}{2} - \mathbf{b}\right) = \frac{s}{2}\mathbf{a} + (1-s)\mathbf{b} + \frac{s}{2}\mathbf{c}$$
5. Equate coefficients of $\mathbf{a}$, $\mathbf{b}$, $\mathbf{c}$ and solve: $1-t = s/2$, $t/2 = 1-s$, $t/2 = s/2$ gives $t = s = 2/3$
6. Substitute back to get common intersection point:

   $$\mathbf{r} = \frac{\mathbf{a} + \mathbf{b} + \mathbf{c}}{3}$$
7. Verify this point also satisfies the parametric equation of CD, hence all three lines pass through the same point and are concurrent.

> **Exam tip:** For concurrency proofs, you only need to show three lines share a common point: use parametric equations for each line and solve for matching scalar parameters.

*Calculator:* allowed

## Common pitfalls

- **Wrong:** Proving collinearity only by showing two vectors are parallel, no mention of a common point
  - Why it fails: Parallel vectors can lie on separate non-intersecting lines, so you cannot confirm collinearity without a shared point
  - Correct: After proving parallelism, explicitly state the common point (e.g. "Vectors PQ and QR share point Q, so P, Q, R are collinear")
- **Wrong:** Flipping the ratio in the section formula, using $\frac{m\mathbf{a} + n\mathbf{b}}{m+n}$ for ratio $AM:MB = m:n$
  - Why it fails: The formula weights the opposite end of the segment to the ratio value, so flipping gives the wrong point
  - Correct: For $AM:MB = m:n$, use $\frac{n\mathbf{a} + m\mathbf{b}}{m+n}$, or verify by calculating distances if you are unsure
- **Wrong:** Omitting the scalar multiple value when proving parallel lines
  - Why it fails: Exam markers require explicit evidence of the scalar multiple relation to award full marks
  - Correct: Write the exact scalar value e.g. $\overrightarrow{CD} = 2\overrightarrow{AB}$ to clearly demonstrate parallelism
- **Wrong:** Using dot product to prove perpendicularity or angles in vector proofs
  - Why it fails: Dot product is explicitly out of scope for the 4PM1 specification for this topic
  - Correct: Restrict all proofs to parallelism, collinearity, concurrency, and ratio division as specified
- **Wrong:** Assuming a ratio is externally divided when it is not specified
  - Why it fails: Only internal ratio division is examined in 4PM1, external division is out of scope
  - Correct: Use the standard internal section formula for all ratio questions, and confirm the point lies between the two endpoints if needed

## Cheatsheet

| Geometric Property | Vector Proof Method | Key Formula/Check |
| --- | --- | --- |
| Parallel lines | Show one vector is non-zero scalar multiple of the other | $\overrightarrow{AB} = \lambda \overrightarrow{CD}, \lambda \neq 0$ |
| Collinear points | Show two vectors between pairs are parallel + share common point | Common point + scalar multiple relation |
| Ratio division (m:n) | Apply internal section formula | $\mathbf{r} = \frac{n\mathbf{a} + m\mathbf{b}}{m+n}$ |
| Concurrent lines | Show all lines pass through the same common intersection point | Common point satisfies parametric equation of all lines |

## What's next

Now that you have mastered vector proofs for geometric properties, you are ready to practice applying these skills to past paper questions, which often combine ratio division and proof tasks in mixed context problems. These vector rules are frequently paired with coordinate geometry questions in the 4PM1 exam, so reinforcing your coordinate geometry foundations will help you tackle longer multi-step questions efficiently. Make sure you memorize the section formula, as it is not provided on the formula sheet and is tested in almost every vector topic exam paper. You should also practice writing clear, step-by-step proofs that explicitly state all required reasoning (e.g. scalar multiples, common points) to avoid losing marks for incomplete working.

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