Position vector

Q: What notation should I use for position vectors in the exam?

Edexcel accepts both bold lowercase letters (e.g. ) or underlined letters (e.g. ) for vectors. You can also use the arrow notation for position vectors relative to origin O, as long as you label vectors clearly.

Edexcel International GCSE Further Pure Mathematics· Section 7D (2016 specification)· 15 min read

1. 1. Definition of Position Vectors Relative to Origin O★☆☆☆☆⏱ 5 min

📘 Definition

Position Vector

\textbf{a} or \bigtriangleup OA

A vector that starts at the fixed origin O and ends at point A. For a point with 2D coordinates $(x_{A}, y_{A})$ , its position vector is written as a column vector: $a = (x_{A} y_{A})$ .

Example:

If point A has coordinates (3, 4) relative to origin O, its position vector is $a = (34)$ .

All position vectors in a single exam question are measured from the same fixed origin (stated at the start of the question). 4PM1 only assesses 2D position vectors, so you will never see a third (z) component in official questions.

📐 Worked Example

Write the position vectors for points P (2, -5) and Q (-1, 0) relative to origin O.

1
For point P, use the x-coordinate as the top component and y-coordinate as the bottom component of the column vector:
2
$\textbf{p} = \bigtriangleup OP = \begin{pmatrix} 2 \\ -5 \end{pmatrix}$
3
For point Q, repeat the process using its coordinates:
4
$\textbf{q} = \bigtriangleup OQ = \begin{pmatrix} -1 \\ 0 \end{pmatrix}$

2. 2. The $\boldsymbol{\bigtriangleup AB = b - a}$ Displacement Rule★★☆☆☆⏱ 6 min

✓ Calculator OK

To find the displacement vector from point A to point B, you subtract the position vector of the starting point (A) from the position vector of the end point (B). This is one of the most frequently tested rules in 4PM1 Section 7.

📘 Definition

Displacement between two points

\bigtriangleup AB

The vector that travels directly from point A to point B, calculated as: $△ A B = △ O B - △ O A = b - a$ , where $a$ and $b$ are the position vectors of A and B respectively.

Example:

If $a = (13)$ and $b = (47)$ , then $△ A B = (4 - 1 7 - 3) = (34)$ .

📐 Worked Example

Point A has position vector $a = (26)$ and point B has position vector $b = (5 - 1)$ relative to origin O. Calculate the vector $△ A B$ .

1
Recall the displacement rule: $△ A B = b - a$ (end point position vector minus start point position vector).
2
Substitute the given position vectors into the rule:
3
$\bigtriangleup AB = \begin{pmatrix} 5 \\ -1 \end{pmatrix} - \begin{pmatrix} 2 \\ 6 \end{pmatrix}$
4
Subtract corresponding components of the vectors:
5
$\bigtriangleup AB = \begin{pmatrix} 5 - 2 \\ -1 - 6 \end{pmatrix} = \begin{pmatrix} 3 \\ -7 \end{pmatrix}$

3. 3. Exam-style Position Vector Problems★★★☆☆⏱ 7 min

✓ Calculator OK

Exam questions for this topic often ask you to find unknown components of position vectors, or relate position vectors to coordinate geometry concepts you already know.

📐 Worked Example

The vector $△ M N = (4 - 2)$ . Point M has position vector $m = (13)$ relative to origin O. Find the coordinates of point N.

1
Start with the displacement rule: $△ M N = n - m$
2
Rearrange to solve for the unknown position vector $n$ :
3
$\textbf{n} = \bigtriangleup MN + \textbf{m}$
4
Substitute the given values and add corresponding components:
5
$\textbf{n} = \begin{pmatrix} 4 \\ -2 \end{pmatrix} + \begin{pmatrix} 1 \\ 3 \end{pmatrix} = \begin{pmatrix} 4 + 1 \\ -2 + 3 \end{pmatrix} = \begin{pmatrix} 5 \\ 1 \end{pmatrix}$
6
Convert the position vector to coordinates: the x-coordinate is the top component, y-coordinate is the bottom component, so N is at (5, 1).

✓ Quick check

If $a = (- 3 2)$ and $b = (1 - 4)$ , what is $△ B A$ ?
- $(4 - 6)$
- $(- 4 6)$
- $(- 2 - 2)$
Reveal answer
1 —
$△ B A = a - b = (- 3 - 1 2 - (- 4)) = (- 4 6)$ . Remember that $△ B A$ is the vector from B to A, so you subtract B's position vector from A's position vector.

4. Common Pitfalls

Wrong move:

Calculating $△ A B$ as $a - b$ instead of $b - a$ .

Why:

Swapping the order of subtraction gives the vector from B to A, which is the negative of the correct answer, leading to lost marks.

Correct move:

Always subtract the position vector of the starting point from the position vector of the end point: end minus start.

Wrong move:

Writing position vectors as $(y x)$ instead of $(x y)$ .

Why:

The top component of a 2D vector corresponds to the horizontal x-coordinate, the bottom to the vertical y-coordinate. Swapping these gives the wrong position.

Correct move:

Copy coordinates directly into the vector: first the x value, then the y value, matching the order of coordinate pairs.

Wrong move:

Using 3D vectors with a z-component for 4PM1 position vector questions.

Why:

The 4PM1 specification explicitly limits position vectors to 2D only, no 3D content is assessed in this topic.

Correct move:

Only use two components for all position vectors in the 4PM1 exam, as official questions will never include a z-coordinate.

Wrong move:

Assuming the origin changes between points in a single question.

Why:

All position vectors in a single question are measured relative to the same fixed origin stated at the start of the question, unless explicitly told otherwise.

Correct move:

Always reference the stated origin O for all position vectors in a question, unless you are given explicit instructions to use a different reference point.

5. Quick Reference Cheatsheet

Concept	Rule	Example
Position vector of A (x,y)	$a = (x y)$	A=(2,5) → $a = (25)$
Displacement $△ A B$	$△ A B = b - a$	$a = (13), b = (41)$ → $△ A B = (3 - 2)$
Find unknown position vector of B	$b = △ A B + a$	$△ A B = (21), a = (30)$ → $b = (51)$

6. Frequently Asked

What notation should I use for position vectors in the exam?

Edexcel accepts both bold lowercase letters (e.g. $a$ ) or underlined letters (e.g. $\b a$ ) for vectors. You can also use the arrow notation $△ O A$ for position vectors relative to origin O, as long as you label vectors clearly.

Going deeper

formula sheetEdexcel 4PM1 Official Specification & Formula SheetRefer to page 34 for given formulae for the exam

What's Next

Now that you have mastered position vectors, you are ready to progress to more advanced vector topics in Edexcel IGCSE Further Pure Math. Next, you will learn about unit vectors, which build directly on the position vector and magnitude concepts you have already covered. You will also use position vectors to solve ratio division problems, such as finding the position vector of a point that divides a line segment in a given ratio. These topics are all core to Section 7 of the 4PM1 specification, and are frequently tested together in multi-mark exam questions. Make sure you practice applying the $△ A B = b - a$ rule regularly to avoid common order-of-subtraction mistakes in your exam.