IGCSE Mathematics (0580) Study Notes: Magnitude of a Vector

Hello Future Mathematicians!

Welcome to the exciting world of vectors! If a vector tells us *where* to go (direction) and *how far* (distance), the concept of **Magnitude** is simply finding that "how far" part—the actual straight-line distance.

In real life, this is incredibly useful. If an aeroplane takes off from the airport, its vector might tell you it flew 3 km East and 4 km North. The magnitude tells you the actual distance between its starting point and current position! Don't worry if this seems tricky at first; we will break it down using a very familiar geometry rule.

1. Quick Review: Understanding Vectors (\( \mathbf{a} \))

Before finding the magnitude (length), we must first remember how we represent a vector in the IGCSE curriculum.

Vector Notation: The Column Vector

A vector is usually written as a column vector, often denoted by a lowercase bold letter like $\mathbf{a}$ or $\mathbf{b}$, or a line segment with an arrow, like $\vec{AB}$.

$$ \mathbf{a} = \begin{pmatrix} x \\ y \end{pmatrix} $$

  • \(x\): Represents the movement in the horizontal direction (East/West).
  • \(y\): Represents the movement in the vertical direction (North/South).

Example: The vector $\begin{pmatrix} 5 \\ -2 \end{pmatrix}$ means move 5 units right (positive x) and 2 units down (negative y).


Key Takeaway: A vector is just a set of instructions for movement: Go $x$ then go $y$.

2. Defining Magnitude (The Size)

The Magnitude of a vector is simply its length or size. It tells us the total distance covered when moving from the start point to the end point of the vector along a straight line.

Analogy: The Road Trip

Imagine you are walking from A to B. The vector $\mathbf{a} = \begin{pmatrix} 3 \\ 4 \end{pmatrix}$ tells you to walk 3 blocks East and then 4 blocks North.

  • Your total walking distance along the grid lines is \(3 + 4 = 7\) blocks.
  • The Magnitude is the shortest, straight-line distance (the "as the crow flies" distance) from A to B.
Notation for Magnitude

We denote the magnitude of a vector using modulus signs (vertical lines), just like absolute value.

  • The magnitude of vector $\mathbf{a}$ is written as \(| \mathbf{a} |\).
  • The magnitude of vector $\vec{AB}$ is written as \(| \vec{AB} |\).

The magnitude is always a positive scalar quantity (a number without direction), because length cannot be negative.

Did You Know? A vector whose magnitude is exactly 1 unit is called a Unit Vector.

3. Calculating Magnitude using Pythagoras' Theorem

This is the most important part! Calculating the magnitude of a two-dimensional vector is a direct application of Pythagoras' theorem.

Visualizing the Triangle

When you draw a vector $\begin{pmatrix} x \\ y \end{pmatrix}$ on a coordinate plane, the horizontal movement ($x$) and the vertical movement ($y$) form the two shorter sides of a right-angled triangle.

The magnitude (the straight-line distance) is the hypotenuse ($c$) of this triangle!

The Formula

Remember Pythagoras' theorem: \(a^2 + b^2 = c^2\).

Applying this to our vector $\begin{pmatrix} x \\ y \end{pmatrix}$:

The length of the hypotenuse, $c$ (which is our magnitude, $| \mathbf{a} |$), is found by:

$$ \text{Magnitude} = | \mathbf{a} | = \sqrt{x^2 + y^2} $$

This formula is provided in the syllabus notes, but it is essential to understand *why* it works—it's just Pythagoras!

Memory Aid: Magnitude = Square, Square, Add, Root!


4. Step-by-Step Examples

Example 1: Positive Components

Calculate the magnitude of the vector $\mathbf{p} = \begin{pmatrix} 8 \\ 6 \end{pmatrix}$.

Step 1: Identify the components (\(x\) and \(y\))

$x = 8$ and $y = 6$.

Step 2: Apply the magnitude formula

$$ | \mathbf{p} | = \sqrt{x^2 + y^2} $$ $$ | \mathbf{p} | = \sqrt{8^2 + 6^2} $$

Step 3: Calculate the squares and add

$$ | \mathbf{p} | = \sqrt{64 + 36} $$ $$ | \mathbf{p} | = \sqrt{100} $$

Step 4: Find the final root

$$ | \mathbf{p} | = 10 \text{ units} $$


Example 2: Including Negative Components

Calculate the magnitude of the vector $\mathbf{q} = \begin{pmatrix} -3 \\ 5 \end{pmatrix}$.

Common Mistake Alert! When calculating, always remember to square the entire negative number. \((-3)^2\) is always positive 9.
Step 1: Identify the components (\(x\) and \(y\))

$x = -3$ and $y = 5$.

Step 2: Apply the magnitude formula (remember the brackets!)

$$ | \mathbf{q} | = \sqrt{(-3)^2 + 5^2} $$

Step 3: Calculate the squares and add

$$ | \mathbf{q} | = \sqrt{9 + 25} $$ $$ | \mathbf{q} | = \sqrt{34} $$

Step 4: Find the final root (and round, if necessary)

Since 34 is not a perfect square, we usually leave the answer as a surd (if asked for the exact value) or round to 3 significant figures.

$$ | \mathbf{q} | = \sqrt{34} \approx 5.83 \text{ units (3 s.f.)} $$


Example 3: Working with Decimals/Fractions

Calculate the magnitude of the vector $\mathbf{v} = \begin{pmatrix} -1.5 \\ -2 \end{pmatrix}$.

Calculation

$$ | \mathbf{v} | = \sqrt{(-1.5)^2 + (-2)^2} $$ $$ | \mathbf{v} | = \sqrt{2.25 + 4} $$ $$ | \mathbf{v} | = \sqrt{6.25} $$ $$ | \mathbf{v} | = 2.5 \text{ units} $$


5. Quick Review: Magnitude of a Vector

The Core Concept in a Box

The magnitude is the length of the vector. It is calculated using the algebraic representation of the vector and the Pythagorean theorem.

Vector: $$ \mathbf{a} = \begin{pmatrix} x \\ y \end{pmatrix} $$

Magnitude Formula: $$ | \mathbf{a} | = \sqrt{x^2 + y^2} $$

Key Rule: Even if $x$ or $y$ are negative, their squares ($x^2$ and $y^2$) must always be positive before adding them together!