Adding and Subtracting Vectors Geometrically
Simple Explanation
To add two vectors geometrically, draw them "tip-to-tail" — place the tail of the second vector at the head (tip) of the first. The sum (resultant) is the single vector from the very first tail to the very last head (the Triangle Law). To subtract a vector, add its negative (reverse) instead.
Why Do We Need It?
Vector addition and subtraction are the single most-used operations in the rest of this chapter — every later idea (geometric proofs, position vectors, ratios) is built directly on top of them.
See It
Vector a from A to B, vector b from B to C drawn tip-to-tail, and the resultant vector a+b drawn directly from A to C
Formula
The Triangle Law of Vector Addition
AB + BC = AC
Placing the tail of a second vector at the head of the first, the sum is the single vector from the very first tail to the very last head.
- AB
- — a vector from point A to point B
- BC
- — a vector from point B to point C, placed tail-to-head with AB
- AC
- — the resultant (sum) vector, from A directly to C
When to use it: Whenever two vectors are added and can be drawn head-to-tail (e.g. combining two journeys, or two forces acting in sequence).
Worked Example
Add two perpendicular vectors using the Triangle Law
A hiker walks vector a: 3 km east, then vector b: 4 km north. Use the Triangle Law to find the magnitude of the resultant displacement a+b.
Why Does This Work?
A vector represents a displacement. Moving by a and then by b, tail-to-head, physically ends at the exact same final position as a single direct displacement straight from the start to that end point — and that direct displacement is, by definition, the sum a+b. The Triangle Law is just this physical fact drawn as a diagram.
Real-Life Example
A pilot combining airspeed and wind velocity
A plane has its own airspeed vector, but the wind blowing past it has a separate velocity vector.
Adding the two vectors (airspeed + wind) using the Triangle Law gives the plane's actual resultant velocity over the ground — essential for accurate flight navigation.
Practice
A boat's own velocity vector is 6 km/h east; the current's velocity vector is 8 km/h north. Find the magnitude of the resultant velocity, in km/h.
MediumCommon mistake
Adding the magnitudes directly (e.g. 3+4=7) instead of combining them geometrically — vectors only add like plain numbers when they point in exactly the same (or exactly opposite) direction.
Quick Review
- Triangle Law: draw vectors tip-to-tail; the resultant runs from the first tail to the last head.
- Subtraction: a−b = a+(−b) — reverse b's direction, then add.
- Perpendicular vectors combine via the Pythagorean theorem for the resultant's magnitude.