Resultant Vectors (Calculations)
Core & Supplement • Syllabus 1.1.7: "Determine, by calculation or graphically, the resultant of two vectors at right angles"
🎯 The Hook: The Compass Problem
You walk 3 km East, then 4 km North. How far are you from where you started?
Your GPS doesn't say "7 km" – it says 5 km at an angle! This is because vectors at right angles combine using Pythagoras, not simple addition.
📐 The Core Concept
When two vectors are at 90° (right angles):
- • The magnitude of the resultant is found using Pythagoras' theorem
- • The direction is found using trigonometry (tan θ = opposite/adjacent)
- • You can solve this graphically (by drawing) or by calculation
🎮 Interactive Vector Adder
Adjust the horizontal (x) and vertical (y) components. Watch the resultant vector and calculations update in real-time.
📊 Real-Time Calculations
🔄 Two Methods, Same Answer
The diagram above shows two graphical methods for finding the resultant:
Head-to-Tail (Triangle Method)
- • Draw the first vector
- • Place the tail of the second vector at the head of the first
- • Draw a line from the start to the end
- • This line is the resultant
Tail-to-Tail (Parallelogram Method)
- • Draw both vectors from the same point
- • Complete the parallelogram with dashed lines
- • Draw the diagonal from the origin
- • This diagonal is the resultant
⚡ Key Point: Both methods give the identical resultant vector. The IGCSE exam may ask you to use either method, so practice both!
⚠️ The "Tug of War" Trap
📝 Mark Scheme Keywords
For full marks in Paper 4, you must:
- • State "Use Pythagoras' theorem" or write the formula R² = x² + y²
- • Show substitution: R² = 3² + 4² = 9 + 16 = 25
- • Write the final answer with units: R = 5 N
- • For direction: "Use tan θ = opposite/adjacent"
- • Show calculation: θ = tan⁻¹(4/3) = 53.1°
✅ Syllabus Checklist (1.1.7)
- ✓Determine the resultant of two vectors at right angles (by calculation using Pythagoras)
- ✓Determine the resultant graphically (using head-to-tail or parallelogram method)
- ✓Calculate direction using trigonometry (tan θ = y/x)
🎯 Exam Success Formula
When vectors are at right angles: ALWAYS use Pythagoras. Never add the magnitudes directly. Show your working, include units, and if drawing graphically, use a ruler and protractor for accuracy.
Keep reading — this lesson marks itself done automatically.