Vector proofs and geometric arguments (IB Math AA HL)

Key ideas

• Choose an origin and name the base vectors at the start, for example OA = a and OC = c.
• Midpoint of PQ: ½(p + q). The point dividing PQ in the ratio m : n is (np + mq) ÷ (m + n).
• Collinear: show AC = k AB for a scalar k, and say that A is a common point. Parallel alone is not enough.
• Perpendicular: show the scalar product is 0. Equal lengths: show |u|² = |v|² using u·u.
• End with a sentence that says what has been proved; the final mark is for the conclusion.

Worked example

(a) OABC is a parallelogram with OA = a and OC = c. Prove that its diagonals bisect each other. (b) Show that A(1, 2, 3), B(3, 5, 4) and C(7, 11, 6) are collinear, and find the ratio AB : BC.

1. (a) OB = a + c, so the midpoint of the diagonal [OB] has position vector ½(a + c).
2. The midpoint of [AC] has position vector ½(a + c) as well, since it is ½(OA + OC).
3. The two midpoints are the same point, so each diagonal passes through the midpoint of the other: they bisect each other.
4. (b) AB = (2, 3, 1) and AC = (6, 9, 3) = 3 AB.
5. AC is a multiple of AB and they share the point A, so A, B and C are collinear. BC = (4, 6, 2) = 2 AB, so AB : BC = 1 : 2.

Answer: (a) Both diagonals have midpoint ½(a + c). (b) AC = 3AB, so the points are collinear, with AB : BC = 1 : 2.

Exam-style HL AA questions on this topic

• Vector Lines questions — 53 questions with mark schemes, plus free worked examples
• All HL AA Geometry and Trigonometry questions by topic

Key formulas for vector proofs and geometric arguments

Vector magnitude

|a⃗| = √(a₁² + a₂² + a₃²)

Vector equation of a line

r⃗ = a⃗ + t d⃗

In the notation of the IB formula booklet. All HL AA formulas · see them used in worked examples

Maths IA ideas that use vector proofs and geometric arguments

• Estimating π by dropping needles: how fast does the estimate improve?
• Patterns in the last digits of powers
• Perfect numbers and Mersenne primes: proving Euclid's rule
• Why does a random game draw the Sierpiński triangle?
• All HL AA IA ideas

Related in Geometry and Trigonometry

• Compound and double-angle formulas
• Trigonometric equations — HL identities in a finite interval
• 3D vectors — dot product, cross product, and applications

Revise Vector proofs and geometric arguments for the exam

• HL AA revision notes — every unit summarised, with the key results to learn
• HL AA past papers — specimen papers and timed practice papers
• Formulas for Geometry and Trigonometry — on the one-page HL AA formula sheet (free PDF), with the ones not in the IB booklet marked
• HL AA formula booklet by topic — every formula in the IB booklet's notation, with a free PDF
• IB Maths prediction papers — full papers in the IB format, with markschemes

TOK link: If mathematics rests on axioms that cannot themselves be proved, in what sense is it certain?

Stretch yourself: original geometry problems at Senior level (ages 16 to 18), with hints and full solutions: Problem S17 · Problem S73 · Problem S16 · Problem S72. All 22 →