Magnitude of a Vector

2026 Syllabus Objectives

By the end of this topic, you should be able to:

  1. Calculate the magnitude of a vector (x/y) as √(x² + y²)
  2. Understand and use modulus signs to denote the magnitude of vectors, e.g. |a| is the magnitude of a; |AB| is the magnitude of AB

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What is Magnitude?

The magnitude of a vector is simply its length. Think of it like measuring how long an arrow is when you draw a vector.

When you have a vector, it has two parts:

  • An x-component (how far it goes left or right)
  • A y-component (how far it goes up or down)

The magnitude tells you the total distance from the start point to the end point of the vector.

Notation: Modulus Signs

We use special symbols called modulus signs to show the magnitude of a vector. These look like vertical bars on either side of the vector name.

Examples of notation:

  • If a vector is called a, its magnitude is written as |a|
  • If a vector goes from point A to point B (written as AB), its magnitude is written as |AB|

Think of the vertical bars like brackets that say "find the length of this vector."

The Magnitude Formula

For any vector written as (xy)\begin{pmatrix} x \\ y \end{pmatrix}, where x is the horizontal component and y is the vertical component, the magnitude is calculated using:

∣vector∣=x2+y2|\text{vector}| = \sqrt{x^2 + y^2}

In words: Take the x-component, square it. Take the y-component, square it. Add these two squares together. Then find the square root of the total.

Why this formula works:

This formula comes from Pythagoras' theorem. Imagine a right-angled triangle where:

  • The base is the x-component
  • The height is the y-component
  • The hypotenuse (longest side) is the vector itself

The magnitude is the length of that hypotenuse, so we use: hypotenuse² = base² + height²

Step-by-Step Method

To find the magnitude of a vector:

Step 1: Identify the x-component and y-component of the vector

Step 2: Square each component (multiply it by itself)

Step 3: Add the two squared values together

Step 4: Find the square root of the sum

Step 5: Simplify if possible

Worked Examples

Example 1: Basic magnitude calculation

Find the magnitude of vector AB⃗=(34)\vec{AB} = \begin{pmatrix} 3 \\ 4 \end{pmatrix}

Solution:

Step 1: x = 3, y = 4

Step 2: x² = 3² = 9 and y² = 4² = 16

Step 3: x² + y² = 9 + 16 = 25

Step 4: ∣AB⃗∣=25=5|\vec{AB}| = \sqrt{25} = 5

Answer: The magnitude is 5

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