Skip to main content
MathidiaMathematics Academy
Vectors and geometric proof
Lesson 50 of 75
Review

Vector proof

Objective. Construct multi-step geometric proofs without coordinates.

  1. 1. Learn
  2. 2. Worked example
  3. 3. Practice
  4. 4. Feedback
  5. 5. Continue

Learn: the key idea

Vector proof builds multi-step geometric arguments, such as showing two lines are parallel or a point is a midpoint, entirely using vector expressions rather than coordinates. Express the relevant sides or diagonals of the shape as combinations of given vectors like a and b, simplify algebraically, then compare results to draw the required conclusion such as parallelism (one vector being a scalar multiple of another). For instance, if PQ = 2a + b and RS = 4a + 2b, then RS = 2PQ, proving PQ is parallel to RS.

Worked example

In triangle OAB, M is the midpoint of OA and N is the midpoint of OB, with OA = a and OB = b. Use vector proof to show MN is parallel to AB.

Diagram of a triangle
  1. 1Write OM = ½a and ON = ½b since M and N are midpoints
  2. 2Find MN = ON − OM = ½b − ½a = ½(b − a)
  3. 3Find AB = OB − OA = b − a
  4. 4Compare: MN = ½(AB), so MN is a scalar multiple of AB, proving they are parallel

Answer: MN = ½AB, so MN is parallel to AB

Concept mastery

Based on your past attempts at this lesson — the question types to practise again come first.

Complete a practice set to see your concept breakdown here.

Practice

Practice happens on its own screen, one question at a time. Answers stay hidden until you submit yours, and you can stop and pick up at the same question later.

Start practising

Common mistake. Students often introduce coordinates or assume specific numerical lengths instead of keeping the proof fully general in terms of the given vectors a and b. Always leave the argument in vector form throughout so the proof applies to any such shape.

Your progress

Tick each step as you finish it. Your exact reading position is saved automatically, so Continue drops you back on the same line.

0 of 5 steps complete

Your place saves automatically as you read.

My dashboard