Vector quantities
Displacement, velocity, force and acceleration.
🎯 Learning Objectives
- define a vector;
- distinguish vectors from scalars;
- use arrow and bold notation;
- identify magnitude and direction;
- recognise real-life vector quantities.
Introduction
This lesson develops Introduction and Vector Notation 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.
What Is a Vector?
A vector is a quantity with both magnitude and direction. It can be represented by an arrow whose length shows size and whose arrowhead shows direction.
A vector answers two questions: “How far?” and “In which direction?”
Vectors and Scalars
Scalar quantities
Distance, speed, mass, temperature and time.
Notation
The vector from A to B may be written as AB⃗, a in bold, or a column vector.
🧭 Interactive Diagram: Vector Explorer
Adjust the horizontal and vertical components. The arrow, magnitude and direction update together.
Vector v = (4, 3), magnitude = 5.00 units, direction = 36.9° above the positive x-axis.
🧠 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.
Worked Examples
Reason through the method
- Identify the known information and what must be found.
- Choose the definition, property or formula that connects them.
- Substitute carefully and show each step.
- Check the notation, units and whether the result is sensible.
Practice Questions
- Define a vector.
- Classify speed and velocity.
- What does the arrowhead represent?
- Give two vector quantities used in real life.
✅ Check Your Work
Show answers and working
- A quantity with magnitude and direction.
- Speed is scalar; velocity is vector.
- Direction.
- Examples include displacement, force, acceleration or velocity.
📘 Remember
Vectors carry both magnitude and direction. Work with horizontal and vertical components separately, and keep signs consistent.
⚠️ 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.