International Mathematics 0607: Study Notes
Chapter 8.3: Magnitude of a Vector
Hello mathematicians! Welcome to the section where we measure the power of a vector.
In this chapter, you will learn how to find the length or size of a vector, known as its magnitude. This is super important because vectors are used everywhere, from calculating the force needed to launch a rocket to figuring out the shortest distance an airplane needs to travel.
1. Understanding Vector Magnitude
1.1 What Exactly is Magnitude?
A vector has two defining characteristics:
1. Direction (which way it is pointing)
2. Magnitude (how big it is or how long it is).
The Magnitude of a vector is simply its length or size. Since it represents a distance, the magnitude must always be a non-negative number (a scalar quantity).
Imagine you walk 3 km East and 4 km North. The vector describes your journey (direction). The magnitude is the shortest straight-line distance from your starting point to your finishing point.
1.2 Review: Column Vectors
In IGCSE Mathematics, we usually deal with vectors in two dimensions, represented in column form:
If we have vector \(\mathbf{a}\), it is written as: $$ \mathbf{a} = \begin{pmatrix} x \\ y \end{pmatrix} $$
Here, \(x\) tells us the horizontal movement (East/West) and \(y\) tells us the vertical movement (North/South).
Key Takeaway: Magnitude is the scalar length of the vector, always positive or zero.
2. The Magnitude Formula: A Right-Angle Connection
Don't worry if finding the length seems complicated—it’s actually something you learned back in Geometry! Calculating the magnitude of a vector is simply an application of the Pythagoras’ Theorem.
2.1 Visualising the Right-Angled Triangle
When you have a vector \(\begin{pmatrix} x \\ y \end{pmatrix}\), the horizontal change (\(x\)) and the vertical change (\(y\)) always form the two shorter sides (legs) of a right-angled triangle.
The magnitude of the vector is the hypotenuse (\(c\)) of that triangle.
2.2 The Formula for Magnitude
If a vector is given by \(\mathbf{a} = \begin{pmatrix} x \\ y \end{pmatrix}\), its magnitude, denoted as \(|\mathbf{a}|\), is calculated using the formula:
$$ \mathbf{Magnitude} = |\mathbf{a}| = \sqrt{x^2 + y^2} $$
Did you know? This is exactly the same formula you use to find the distance between two points in coordinate geometry! This shows how interconnected different topics in maths are.
3. Step-by-Step Calculation
Let’s walk through an example to see how easy this formula is to use.
Example: Finding the Magnitude of Vector \( \mathbf{v} \)
Find the magnitude of the vector \(\mathbf{v} = \begin{pmatrix} 5 \\ 12 \end{pmatrix}\).
Step 1: Identify the components \(x\) and \(y\).
In this case, \(x = 5\) and \(y = 12\).
Step 2: Substitute the values into the magnitude formula.
$$ |\mathbf{v}| = \sqrt{x^2 + y^2} $$ $$ |\mathbf{v}| = \sqrt{5^2 + 12^2} $$
Step 3: Square the components.
$$ |\mathbf{v}| = \sqrt{25 + 144} $$
Step 4: Add the squared results.
$$ |\mathbf{v}| = \sqrt{169} $$
Step 5: Calculate the final square root.
$$ |\mathbf{v}| = 13 $$
The magnitude of vector \(\mathbf{v}\) is 13 units.
3.1 Dealing with Negative Components
Don't worry if your vector has negative components (meaning it moves left or down). Because we square \(x\) and \(y\), the result is always positive!
Example with Negatives: Find the magnitude of \(\mathbf{w} = \begin{pmatrix} -3 \\ 4 \end{pmatrix}\).
$$ |\mathbf{w}| = \sqrt{(-3)^2 + 4^2} $$
Watch out! Always place negative numbers in brackets when squaring them.
$$ |\mathbf{w}| = \sqrt{9 + 16} $$ $$ |\mathbf{w}| = \sqrt{25} $$ $$ |\mathbf{w}| = 5 $$
Common Mistake to Avoid:
Forgetting that \((-3)^2\) is positive 9. Students sometimes mistakenly calculate \(-3^2 = -9\), which will lead to an incorrect answer (and mathematically impossible if the final result under the square root is negative!).
4. Notation and Final Answer Presentation
4.1 Modulus Signs: The Mathematical Rulers
In maths, when we want to show that we are finding the magnitude (the size) of something, we use modulus signs (vertical bars: \(| \dots |\)).
- If the vector is written as a bold lowercase letter, e.g., \(\mathbf{a}\), the magnitude is written as \(|\mathbf{a}|\).
- If the vector is represented by two points, e.g., \(\vec{AB}\), the magnitude is written as \(|\vec{AB}|\).
Remember this Mnemonic:
Magnitude is like using a Measurement Modulus.
4.2 Accuracy and Surd Form
Your final answer for magnitude should usually be given in one of two ways:
- Exact Form (Surd Form): If the result under the square root is not a perfect square, you may leave the answer in simplified surd form if required, e.g., \(\sqrt{18} = 3\sqrt{2}\).
- Decimal Form: If the question does not ask for the exact value, provide your answer rounded to 3 significant figures (or the degree of accuracy specified by the question).
Example: If \(\mathbf{u} = \begin{pmatrix} 1 \\ 3 \end{pmatrix}\)
$$ |\mathbf{u}| = \sqrt{1^2 + 3^2} = \sqrt{1 + 9} = \sqrt{10} $$
Exact Answer: \(\sqrt{10}\)
3 s.f. Answer: \(3.16\)
Quick Review Box
Definition: Magnitude is the length of the vector.
Formula: \(|\mathbf{a}| = \sqrt{x^2 + y^2}\)
Key Step: Always square \(x\) and \(y\) (negatives become positive!).
Notation: Use modulus signs \(|\mathbf{a}|\).
Great job! You now know how to calculate the size of any 2D vector. This tool is fundamental as you continue to work with vectors in mechanics and further mathematics.