IB Maths AA HL Topic 3 — Geometry & Trigonometry Paper 1 & 2 ~7 min read HL only

Geometric Proof with Vectors

Vectors give you a clean toolkit for geometry: parallel means scalar multiple; perpendicular means dot product zero; equal length means equal magnitude; same point means equal position vectors. Combine these to prove shapes, find midpoints, and show points are collinear.

📘 What you need to know

Parallel: v = kw (or v × w = 0).
Perpendicular: v · w = 0.
Equal length: |v| = |w|.
Equal vectors: same components → same length AND same direction.
Midpoint of AB has position vector 12(a + b).
Point dividing AB in ratio p:q: AX = pp+q AB.
Collinear: A, B, C on the same line ⟺ any two of AB, AC, BC are parallel.

The proof toolkit

To prove…	Show that…
two vectors are parallel	v = kw, OR v × w = 0
two vectors are perpendicular	v · w = 0
two vectors are equal length	\\|v\\| = \\|w\\|
two segments are equal & parallel	v = w, OR v = −w
three points are collinear	two of AB, AC, BC are parallel
M is the midpoint of AB	m = ½(a + b), or AM = ½AB

Identifying quadrilaterals

Shape	Vector conditions to prove it
Parallelogram	opposite sides equal: AB = DC AND AD = BC
Rectangle	parallelogram + adjacent sides perpendicular (AB · AD = 0)
Rhombus	parallelogram + adjacent sides equal length (\\|AB\\| = \\|AD\\|)
Square	parallelogram + adjacent sides perpendicular AND equal length
Trapezium	only one pair of opposite sides parallel

For ABCD going around the shape, “opposite sides” means AB ↔ DC and AD ↔ BC. Watch the labelling direction: AB and DC point the same way around the parallelogram.

Midpoints and division of a line

Midpoint of AB m = 12(a + b)

Point dividing AB in ratio p:q AX = pp+q AB

The midpoint formula is the special case p = q = 1, giving 12(a + b). For an unequal split, p:q means X is p/(p+q) of the way from A to B.

🧭 Recipe — prove a quadrilateral ABCD is a parallelogram

Compute the four side vectors: AB, BC, CD, DA.
Check opposite-side equality: AB = DC (or equivalently AB = −CD).
Check the other pair: AD = BC (or AD = −DA reversed).
If both pairs match → parallelogram.
For rectangle/rhombus/square, add the perpendicular and/or equal-length checks on adjacent sides.

Worked examples

WE 1

Find the midpoint of a line segment

Find the midpoint M of the line segment joining A(2, −1, 4) and B(8, 3, −2).

Apply m = ½(a + b) m = ½((2, −1, 4) + (8, 3, −2)) = ½(10, 2, 2) M(5, 1, 1) just average each coordinate — that’s all the midpoint formula does

WE 2

Find a point dividing a segment in a given ratio

The point P lies on segment AB and divides it in the ratio 2 : 1, where A(1, 0, 5) and B(7, 6, −1). Find the coordinates of P.

Step 1: AB = b − a AB = (6, 6, −6) Step 2: AP = (2/3)AB (since p:q = 2:1) AP = (2/3)(6, 6, −6) = (4, 4, −4) Step 3: P = A + AP P = (1+4, 0+4, 5−4) P(5, 4, 1) P is 2/3 of the way from A to B (closer to B)

WE 3

Show three points are collinear

The points A, B, and C have coordinates (2, 0, 3), (5, −1, 1), and (11, −3, −3) respectively. Show that A, B, and C are collinear.

Step 1: Compute AB and AC AB = B − A = (3, −1, −2) AC = C − A = (9, −3, −6) Step 2: Check ratios 9/3 = 3; −3/−1 = 3; −6/−2 = 3 Step 3: AC = 3 AB → AB and AC parallel; share point A A, B, C are collinear parallel displacements + shared point = on the same line

WE 4

Prove a quadrilateral is a parallelogram

Use vectors to prove that ABCD with A(1, 2, −1), B(4, 5, 1), C(6, 8, 5), D(3, 5, 3) is a parallelogram.

Step 1: Compute the four side vectors AB = B − A = (3, 3, 2) DC = C − D = (3, 3, 2) AD = D − A = (2, 3, 4) BC = C − B = (2, 3, 4) Step 2: Compare opposite sides AB = DC ✓ and AD = BC ✓ ABCD is a parallelogram two pairs of opposite sides equal as vectors → parallelogram

WE 5

Prove a quadrilateral is a rectangle

Use vectors to prove that ABCD with A(0, 0, 0), B(1, 2, 2), C(3, 1, 2), D(2, −1, 0) is a rectangle but not a square.

Step 1: Show it’s a parallelogram AB = (1, 2, 2); DC = C − D = (1, 2, 2) ✓ AD = (2, −1, 0); BC = (2, −1, 0) ✓ Step 2: Show adjacent sides perpendicular AB · AD = (1)(2) + (2)(−1) + (2)(0) = 0 ✓ Step 3: Show side lengths differ (not square) |AB| = √(1 + 4 + 4) = 3 |AD| = √(4 + 1 + 0) = √5 ABCD is a rectangle (not a square, since 3 ≠ √5) rectangle = parallelogram + perpendicular adjacent sides; equal lengths would make it a square

WE 6

Find an endpoint given a midpoint

The point M(3, −1, 4) is the midpoint of segment AB. Given that A has coordinates (1, 2, −3), find the coordinates of B.

Step 1: Use m = ½(a + b) → b = 2m − a b = 2(3, −1, 4) − (1, 2, −3) = (6, −2, 8) − (1, 2, −3) = (5, −4, 11) B(5, −4, 11) Sanity check: midpoint of A and B ½((1, 2, −3) + (5, −4, 11)) = ½(6, −2, 8) = (3, −1, 4) ✓

💡 Top tips

Always sketch the figure before starting any vector proof.
Pick the simplest vertex to start vectors from — usually the one with smallest or zero coordinates.
For ABCD parallelogram, check AB = DC, NOT AB = CD. Going around the shape, opposite sides point the same way.
For rectangles, the dot product step is critical — without perpendicularity it could just be a parallelogram or rhombus.
Always sanity-check midpoints by averaging the two endpoint coordinates.

⚠ Common mistakes

Comparing AB to CD instead of DC. They’re negatives of each other for a parallelogram.
Stopping at “parallelogram” when the question asks for “rectangle”. Add the perpendicular check.
Confusing “equal” with “parallel”. Equal vectors are also parallel; parallel vectors aren’t always equal.
Forgetting the sanity check on midpoint or ratio answers.
Mixing up the ratio direction: AX = (p/(p+q))AB, so the larger fraction goes with the closer endpoint.

That closes Vector Properties. Up next: Vector Equations of Lines. A line in 3D is described as r = a + λd — a starting point plus a direction vector multiplied by a parameter. The same idea opens up parallel lines, intersection problems, and shortest-distance calculations.

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