📘 Remember
Vectors carry both magnitude and direction. Work with horizontal and vertical components separately, and keep signs consistent.
🎯 Learning Objectives
- calculate vector magnitude;
- connect magnitude with Pythagoras;
- find unit vectors;
- interpret distance from components.
Introduction
This lesson develops Magnitude of a Vector from meaning and patterns before moving to calculations. Look for what the quantities, shapes or symbols represent, then connect that idea to the method.
Key Notes
Read the definitions and key facts below carefully. Each rule is most useful when you can explain why it fits the situation, not simply recall it.
Magnitude Formula
For v=(x,y), |v|=√(x²+y²).
Worked Example
For v=(6,8): |v|=√(36+64)=√100=10.
🧭 Interactive Diagram: Magnitude and Pythagoras
Change the components and see the right triangle behind the magnitude formula.
|v| = √(3² + 4²) = 5.00 units.
🧠 Let’s Think Together
Before calculating, identify the information given, the result required and the mathematical relationship that connects them. Predict whether the answer should be larger, smaller or unchanged, then use that prediction to check the result.
Practice Questions
- Find |(3,4)|.
- Find |(−5,12)|.
- Find the magnitude of (7,−7).
- Why does a negative component not make magnitude negative?
✅ Check Your Work
Show answers and working
- √(9+16)=5.
- √(25+144)=13.
- √(49+49)=√98=7√2≈9.90.
- Components are squared, and magnitude represents a non-negative length.
⚠️ Common Mistakes
Do not confuse a point with its position vector. For AB, always calculate B − A, not A − B.
⭐ Exam Tips
Sketch arrows, label directions clearly, show component working and check whether the final direction agrees with the diagram.
🌍 Did You Know?
Vectors are used in navigation, animation, engineering, robotics, weather modelling and computer games.
Common Mistakes
- Choosing a rule before identifying what the question describes.
- Skipping working or changing notation part-way through a solution.
- Accepting an answer without checking its size, sign or units.
Exam Focus
Show the mathematical relationship you are using, keep notation consistent and give a clear final answer. Use a quick estimate, diagram or inverse operation to check that the result is reasonable.
Summary
- Understand the meaning of each idea before selecting a method.
- Use definitions, properties and notation consistently.
- Show working and check that the final result is sensible.