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Maths

Vectors in Geometry

Overview

Students apply vector notation and algebra to prove geometric properties — a powerful method that replaces coordinate geometry for many problems and introduces the elegance of vector proof.

Learning Objective
Students use vectors to prove geometric results including collinearity and midpoint relationships, express position vectors, and apply vector methods to solve geometric problems.

Resources needed

  • Mini whiteboards
  • Diagrams on board

Lesson stages

0 / 7 done
  1. 1 Recap: a vector has magnitude and direction. Express as a column vector, using bold letters (a), or with arrow (AB⃗). AB⃗ = B − A as position vectors. BA⃗ = −AB⃗. Add: a + b (tip to tail). Scalar multiple: 3a (same direction, three times as long).
  2. 2 If O is the origin, OA⃗ = a, OB⃗ = b. AB⃗ = OB⃗ − OA⃗ = b − a. Find: if a = (2,3) and b = (5,1), find AB⃗ = (3,−2). Magnitude: |AB| = √(9+4) = √13. Students practice finding vectors between given position vectors.
  3. 3 The midpoint M of AB has position vector OM⃗ = ½(a+b). Proof: OM⃗ = OA⃗ + AM⃗ = a + ½AB⃗ = a + ½(b−a) = ½a + ½b = ½(a+b). Students find midpoints of several line segments using vectors.
  4. 4 Three points are collinear if one vector between them is a scalar multiple of another. Example: A, B, C with OA=a, OB=3a−b, OC=5a−2b. AB⃗=2a−b. AC⃗=4a−2b=2(2a−b)=2·AB⃗. Since AC⃗ = 2AB⃗ and they share point A, the points are collinear.
  5. 5 P divides AB in ratio 2:3. OP⃗ = OA⃗ + 2/5 AB⃗ = a + 2/5(b−a) = 3/5 a + 2/5 b. Students find position vectors for points dividing segments in various ratios.
  6. 6 In triangle OAB, M is midpoint of OA, N is midpoint of OB. Prove MN is parallel to AB and half its length. OM⃗ = ½a, ON⃗ = ½b. MN⃗ = ON⃗ − OM⃗ = ½b − ½a = ½(b−a) = ½AB⃗. So MN ∥ AB and |MN| = ½|AB|. QED.
  7. 7 In quadrilateral OABC, M, N, P, Q are midpoints of OA, AB, BC, OC respectively. Prove that MNPQ is a parallelogram. Students work in pairs through this multi-step proof.

Tap a step to mark it as done.

Variations

  • Extend to 3D vectors
  • Apply vectors to forces in mechanics
  • Introduce the dot product for angle calculations
More information

Display: position vector, resultant, scalar multiple, collinear, parallel, midpoint, ratio. Notation: bold for vectors (a), arrow for line vectors (AB⃗).

Focus on 2D column vectors only. Provide diagrams for all problems. Scaffold the proof structure with given opening lines.

Do students express AB⃗ as b − a (not a − b)? Is scalar multiplication applied correctly? In collinearity proofs, do they identify a common point AND a scalar multiple relationship?

All diagrams drawn on plain paper. Mini whiteboards for calculations. No specialised resources needed.

Students often write AB⃗ = a − b instead of b − a. Reinforce: AB⃗ = 'end minus start' = OB⃗ − OA⃗ = b − a.

Activity details
Secondary Challenging 50 minutes No prep Solo, Pairs Indoor
Topics
Vectors Geometry Proof Midpoints Parallel