Vector arithmetic and geometric proofs
Vectors are powerful tools for describing positions and proving geometric properties. GCSE Higher requires you to operate on vectors and use them to construct proofs.
Vector addition (head-to-tail)
To add a + b: place tail of b at head of a. The sum is the arrow from start to finish.
In components: (a₁ / a₂) + (b₁ / b₂) = (a₁ + b₁ / a₂ + b₂).
Vector subtraction
a − b = a + (−b). Geometrically: head-to-tail with the reversed b.
In components: (a₁ / a₂) − (b₁ / b₂) = (a₁ − b₁ / a₂ − b₂).
Scalar multiplication
ka scales the vector by factor k:
- k > 1: same direction, longer.
- 0 < k < 1: same direction, shorter.
- k < 0: reversed direction.
In components: k(a₁ / a₂) = (ka₁ / ka₂).
Position vectors
If A has coordinates (x, y), then a = ⃗OA = (x / y) is the position vector of A.
⃗AB = b − a (head minus tail).
Midpoint
If M is the midpoint of AB, then m = ½(a + b).
Geometric proofs using vectors
A common GCSE proof structure:
Show that PQ is parallel to RS, where P, Q, R, S are defined in terms of vectors.
If ⃗PQ = k × ⃗RS for some scalar k, then PQ is parallel to RS.
If additionally |k| = 1, they're the same length too (so PQRS is a parallelogram, etc.).
✦Worked example
In a triangle OAB, ⃗OA = a and ⃗OB = b. M is the midpoint of AB. Find ⃗OM.
- ⃗OM = ⃗OA + ⃗AM = a + ½(⃗AB) = a + ½(b − a) = a + ½b − ½a = ½a + ½b.
So ⃗OM = ½(a + b) — the midpoint formula.
✦Worked example— Worked example — proving parallel
OABC is a parallelogram. ⃗OA = a, ⃗OC = c. M is the midpoint of OB. Show that M is also on AC.
- ⃗OM = ½ ⃗OB. ⃗OB = ⃗OA + ⃗AB = a + c (in parallelogram).
- So ⃗OM = ½(a + c).
- Midpoint of AC is also ½(a + c) (using midpoint formula).
- Therefore the midpoints coincide → diagonals bisect each other in parallelogram. ✓
⚠Common mistakes
- Direction reversal — ⃗AB = −⃗BA.
- Forgetting head minus tail — ⃗AB = position of B − position of A.
- Treating parallel as same vector — parallel means proportional (one is a scalar multiple of the other).
- Missing brackets in scalar multiplication, e.g. ½(a + b) ≠ ½a + b.
- Inconsistent labelling — pick a convention and stick to it.
➜Try this— Quick check
If a = (3 / 2) and b = (1 / 5), find 2a − b.
- 2(3 / 2) − (1 / 5) = (6 / 4) − (1 / 5) = (5 / −1).
AI-generated · claude-opus-4-7 · v3-deep-geometry