📘 Remember
Vectors carry both magnitude and direction. Work with horizontal and vertical components separately, and keep signs consistent.
🎯 Learning Objectives
- write position vectors;
- find vectors joining points;
- find coordinates from vector movements;
- use midpoint and division ideas.
Introduction
This lesson develops Position Vectors and Points 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.
Position Vectors
The position vector OP⃗ gives the location of P relative to the origin O.
Finding a New Point
If A=(2,−1) and AB⃗=(4,3), then B=A+AB⃗=(6,2).
Midpoint Using Vectors
If OA⃗=a and OB⃗=b, then the midpoint M has OM⃗=(a+b)/2.
🧭 Interactive Diagram: Position Vector and Displacement
Click anywhere inside the coordinate grid to place point P. The position vector OP updates instantly.
OP = (3, 3). Click the grid to move P.
🧠 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
- Write the position vector of P(−3,5).
- A=(1,2), AB⃗=(5,−4). Find B.
- A=(2,6), B=(8,−2). Find midpoint M.
✅ Check Your Work
Show answers and working
- OP⃗=(−3,5).
- B=(1+5,2−4)=(6,−2).
- M=((2+8)/2,(6−2)/2)=(5,2).
⚠️ 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.