# Magnitude of a vector

> Edexcel International GCSE Further Pure Mathematics · 4PM1
> Source: https://www.owlsprep.com/study/edexcel-igcse-further-pure-math-s7-magnitude-of-a-vector/

This guide teaches you to calculate the magnitude (modulus) of 2D vectors in i,j component form for your Edexcel IGCSE Further Pure Math (4PM1) exam, including exact surd answer rules and common exam pitfalls.

**Prerequisites:** [2D vectors in i,j component form](https://www.owlsprep.com/study/edexcel-igcse-further-pure-math-s7-vector-component-form/)

## Learning objectives

- Calculate the magnitude of a 2D vector given in i,j component form
- Leave magnitude answers in fully simplified exact surd form where required
- Apply magnitude calculation to simple exam-style problems involving vector addition/subtraction

## Introduction to Vector Magnitude

**Magnitude of a 2D vector** — The length of the vector, calculated as the positive square root of the sum of the squares of its x and y components.

*Notation:* \vert a\mathbf{i} + b\mathbf{j} \vert

*Example:* For vector 3i + 4j, magnitude = √(3² + 4²) = 5

Magnitude is a scalar quantity, meaning it has size but no direction. Unlike vector components, it can never be negative, since we take the positive square root when calculating it. For Edexcel 4PM1 exams, you will only be required to calculate magnitudes for 2D vectors, with no 3D vectors included in the specification.

**Worked example:** Find the magnitude of vector v = 2i + 3j. Give your answer in exact form.

1. Identify the x and y components of the vector: a = 2 (i component), b = 3 (j component)
2. $$\vert \mathbf{v} \vert = \sqrt{a^2 + b^2} = \sqrt{2^2 + 3^2}$$
3. Calculate the sum of squares: 4 + 9 = 13
4. Take the positive square root, leave as a surd since exact form is required: |v| = √13

> **Exam tip:** Always show the full surd step before giving a decimal approximation, even if the question asks for a decimal value, to earn method marks.

*Calculator:* allowed

## Magnitude of Vectors with Negative Components

Vectors can have negative components if they point in the negative x or y direction. When calculating magnitude, squaring these negative components will always give a positive value, so you do not need to adjust the sign before squaring.

**Worked example:** Calculate the magnitude of vector w = -4i + 2j, giving your answer in fully simplified surd form.

1. Extract components: a = -4 (i component), b = 2 (j component)
2. $$\vert \mathbf{w} \vert = \sqrt{(-4)^2 + (2)^2}$$
3. Calculate squares: (-4)² = 16, 2² = 4, sum = 20
4. Simplify the surd: √20 = √(4×5) = 2√5

> **tip**
>
> You do not need to take the absolute value of components before squaring, since squaring a negative number automatically produces a positive result.

**Check your understanding**

Test your understanding before moving on:

1. What is the magnitude of vector -i - 3j?

   - √10
   - -√10
   - 10
   - 4

   *Answer:* √10

   *Why:* Correct: √((-1)² + (-3)²) = √(1 + 9) = √10. Magnitude is always positive, so option B is invalid.

*Calculator:* allowed

## Exam-Style Resultant Vector Magnitude Problems

In Edexcel 4PM1 exams, magnitude questions are often combined with simple vector addition or subtraction. Remember to first calculate the resultant vector before finding its magnitude, as the magnitude of a sum is not equal to the sum of individual magnitudes.

**Worked example:** Given vectors p = 3i - 2j and q = -i + 5j, find the exact magnitude of p + q.

1. First calculate the resultant vector p + q: add corresponding i and j components
2. $$\mathbf{p} + \mathbf{q} = (3 - 1)\mathbf{i} + (-2 + 5)\mathbf{j} = 2\mathbf{i} + 3\mathbf{j}$$
3. Apply the magnitude formula to the resultant vector:
4. $$\vert \mathbf{p} + \mathbf{q} \vert = \sqrt{2^2 + 3^2} = \sqrt{13}$$

> **Exam tip:** If you calculate magnitudes of individual vectors before adding them, you will get an incorrect result, so always compute the resultant vector first.

*Calculator:* allowed

## Common pitfalls

- **Wrong:** Taking the negative square root when calculating magnitude
  - Why it fails: Magnitude is a measure of length, which cannot be negative
  - Correct: Always take the positive value of the square root for magnitude calculations.
- **Wrong:** Forgetting to square negative vector components before summing
  - Why it fails: Negative components squared give positive values, so omitting the square leads to incorrect negative or too small sums
  - Correct: Square every component, regardless of sign, before adding them together.
- **Wrong:** Leaving surd answers unsimplified e.g. writing √20 instead of 2√5
  - Why it fails: Edexcel examiners require fully simplified exact form unless stated otherwise
  - Correct: Simplify all surd answers by factoring out perfect squares from under the root.
- **Wrong:** Calculating magnitude of individual vectors before adding them for resultant magnitude questions
  - Why it fails: Magnitude of a sum is not equal to the sum of magnitudes (triangle inequality)
  - Correct: First calculate the resultant vector by adding/subtracting components, then find its magnitude.

## Cheatsheet

| Vector Form | Magnitude Formula | Key Rule |
| --- | --- | --- |
| a\mathbf{i} + b\mathbf{j} | √(a² + b²) | Always positive, scalar value |
| \mathbf{v} = x\mathbf{i} + y\mathbf{j} | \vert \mathbf{v} \vert = √(x² + y²) | Leave as simplified surd unless decimal is required |
| Resultant vector \mathbf{u} + \mathbf{v} | First compute \mathbf{u} + \mathbf{v}, then apply formula | Do not add magnitudes separately |

## What's next

Now that you can calculate the magnitude of 2D vectors, you are ready to progress to related vector topics in the Edexcel IGCSE Further Pure Math (4PM1) syllabus. Next, you will learn about position vectors, which describe the location of a point relative to the origin, and use magnitude calculations to find distances between points in coordinate space. You will also later apply this magnitude formula to find unit vectors, which are vectors of length 1 used in more advanced vector operations. Mastering vector magnitude is a foundational skill for all subsequent vector work in your course, and will also be useful if you progress to A-Level Mathematics in the future. Make sure to practice plenty of exam-style questions to reinforce your understanding and avoid the common pitfalls outlined in this guide.

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