Magnitude of a vector

Edexcel International GCSE Further Pure Mathematics· Specification S7, 7C· 15 min read

1. Introduction to Vector Magnitude★☆☆☆☆⏱ 4 min

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📘 Definition

Magnitude of a 2D vector

\vert a\mathbf{i} + b\mathbf{j} \vert

The length of the vector, calculated as the positive square root of the sum of the squares of its x and y components.

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.

2. Magnitude of Vectors with Negative Components★★☆☆☆⏱ 5 min

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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

✓ Quick check

Test your understanding before moving on:

What is the magnitude of vector -i - 3j?
- √10
- -√10
- 10
- 4

3. Exam-Style Resultant Vector Magnitude Problems★★★☆☆⏱ 6 min

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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.

4. Common Pitfalls

Wrong move:

Taking the negative square root when calculating magnitude

Why:

Magnitude is a measure of length, which cannot be negative

Correct move:

Always take the positive value of the square root for magnitude calculations.

Wrong move:

Forgetting to square negative vector components before summing

Why:

Negative components squared give positive values, so omitting the square leads to incorrect negative or too small sums

Correct move:

Square every component, regardless of sign, before adding them together.

Wrong move:

Leaving surd answers unsimplified e.g. writing √20 instead of 2√5

Why:

Edexcel examiners require fully simplified exact form unless stated otherwise

Correct move:

Simplify all surd answers by factoring out perfect squares from under the root.

Wrong move:

Calculating magnitude of individual vectors before adding them for resultant magnitude questions

Why:

Magnitude of a sum is not equal to the sum of magnitudes (triangle inequality)

Correct move:

First calculate the resultant vector by adding/subtracting components, then find its magnitude.

5. Quick Reference 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

6. Frequently Asked

Do I need to simplify surd answers for vector magnitude?

Yes, unless the question explicitly asks for a decimal approximation, always leave magnitude values in fully simplified exact surd form for Edexcel 4PM1 exams to earn full marks.

Can the magnitude of a vector be negative?

No, magnitude is a measure of length, so it is always a non-negative scalar value, even if one or both of the vector's components are negative.

Going deeper

syllabusEdexcel 4PM1 Official SpecificationRefer to section S7 for full vector topic requirements

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.