Use of vectors to prove geometrical properties

Edexcel International GCSE Further Pure Mathematics· 7F· 25 min read

1. Proving Parallel Lines Using Vectors★★☆☆☆⏱ 6 min

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📘 Definition

Parallel Vectors

Two vectors are parallel if one is a non-zero scalar multiple of the other. If $A B = λ C D$ for some non-zero constant $λ \in R$ , then line AB is parallel to line CD.

📐 Worked Example

Points A, B, C, D have position vectors $a = (13)$ , $b = (57)$ , $c = (2 - 2)$ , $d = (84)$ . Prove line AB is parallel to line CD.

1
Calculate vector $A B$ 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 $C D$ :
$\overrightarrow{CD} = \mathbf{d} - \mathbf{c} = \begin{pmatrix}8-2 \\4 - (-2)\end{pmatrix} = \begin{pmatrix}6 \\6\end{pmatrix}$
3
Identify the scalar multiple relation between the two vectors:
$\overrightarrow{CD} = 1.5 \times \overrightarrow{AB}$
4
Since $C D$ is a non-zero scalar multiple of $A B$ , 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.

2. Proving Collinearity of Three Points★★★☆☆⏱ 7 min

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📘 Definition

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 $p = (21)$ , $q = (63)$ , $r = (126)$ . Prove P, Q, R are collinear.

1
Calculate vector $P Q$ :
$\overrightarrow{PQ} = \mathbf{q} - \mathbf{p} = \begin{pmatrix}6-2 \\3-1\end{pmatrix} = \begin{pmatrix}4 \\2\end{pmatrix}$
2
Calculate vector $QR$ :
$\overrightarrow{QR} = \mathbf{r} - \mathbf{q} = \begin{pmatrix}12-6 \\6-3\end{pmatrix} = \begin{pmatrix}6 \\3\end{pmatrix}$
3
Confirm parallelism:
$\overrightarrow{QR} = 1.5 \times \overrightarrow{PQ}$
4
State the common point: vectors $P Q$ and $QR$ 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.

3. Position Vector of a Point Dividing a Line in Ratio m:n★★★☆☆⏱ 7 min

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📘 Definition

Section Formula (Internal Ratio Division)

If point M divides the line segment joining A (position vector $a$ ) and B (position vector $b$ ) in the ratio $m : n$ ( $A M : M B = m : n$ ), then the position vector of M is $m = \frac{n a + m b}{m + n}$ .

📐 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: $m = 2$ , $n = 1$ , $a = (15)$ , $b = (72)$
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 $a$ is the second value in the ratio, and the multiplier for $b$ is the first value.

4. Proving Concurrency of Lines★★★★☆⏱ 7 min

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📘 Definition

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 $a$ , $b$ , $c$ 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 $a$ , $b$ , $c$ 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.

5. Common Pitfalls

Wrong move:

Proving collinearity only by showing two vectors are parallel, no mention of a common point

Why:

Parallel vectors can lie on separate non-intersecting lines, so you cannot confirm collinearity without a shared point

Correct move:

After proving parallelism, explicitly state the common point (e.g. "Vectors PQ and QR share point Q, so P, Q, R are collinear")

Wrong move:

Flipping the ratio in the section formula, using $\frac{m a + n b}{m + n}$ for ratio $A M : M B = m : n$

Why:

The formula weights the opposite end of the segment to the ratio value, so flipping gives the wrong point

Correct move:

For $A M : M B = m : n$ , use $\frac{n a + m b}{m + n}$ , or verify by calculating distances if you are unsure

Wrong move:

Omitting the scalar multiple value when proving parallel lines

Why:

Exam markers require explicit evidence of the scalar multiple relation to award full marks

Correct move:

Write the exact scalar value e.g. $C D = 2 A B$ to clearly demonstrate parallelism

Wrong move:

Using dot product to prove perpendicularity or angles in vector proofs

Why:

Dot product is explicitly out of scope for the 4PM1 specification for this topic

Correct move:

Restrict all proofs to parallelism, collinearity, concurrency, and ratio division as specified

Wrong move:

Assuming a ratio is externally divided when it is not specified

Why:

Only internal ratio division is examined in 4PM1, external division is out of scope

Correct move:

Use the standard internal section formula for all ratio questions, and confirm the point lies between the two endpoints if needed

6. Quick Reference Cheatsheet

Geometric Property	Vector Proof Method	Key Formula/Check
Parallel lines	Show one vector is non-zero scalar multiple of the other	$A B = λ C D, λ \neq = 0$
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	$r = \frac{n a + m b}{m + n}$
Concurrent lines	Show all lines pass through the same common intersection point	Common point satisfies parametric equation of all lines

7. Frequently Asked

Do I need to explicitly state the scalar multiple for parallel line proofs?

Yes, Edexcel exam markers require you to write the exact scalar multiple value (e.g. $C D = 1.5 A B$ ) to get full marks for parallelism proofs, do not only state the lines are parallel.

Can I use the dot product for vector geometric proofs in 4PM1?

No, the dot product is explicitly out of scope for this specification. All proofs must only use scalar multiples, position vector arithmetic, collinearity and concurrency rules.

What is the difference between parallel lines and collinear points?

Parallel lines do not need to share a common point, while collinear points require two parallel vectors and a shared common point to prove they lie on the same straight line.

Going deeper

formula_sheetEdexcel 4PM1 Official Formula SheetSection formula is not provided, must be memorized

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.