Vector Proofs in Geometry
Key Terms
- Collinear points
- A, B, C are collinear if and only if AB = kAC for some scalar k (i.e. the vectors are parallel and share a common point).
- Midpoint vector
- The midpoint M of AB has position vector m = ½(a + b).
- Parallelogram condition
- ABCD is a parallelogram if and only if AB = DC (opposite sides are equal and parallel).
- Centroid
- The intersection of medians; position vector g = ⅓(a + b + c).
- Proof strategy
- Express all vectors in terms of two independent basis vectors; manipulate algebraically to reach the desired result.
- Equal vectors
- Two vectors are equal if they have the same magnitude AND direction; they need not be the same arrow in space.
Setting Up Vector Proofs
Step 1: Assign position vectors. Let O be the origin; points A, B, C have position vectors a, b, c.
Step 2: Express all other points using these base vectors.
Key relationships:
• Midpoint of AB: OM = ½(a + b)
• Vector from A to B: AB = b − a
• Parallel condition: XY = k · UV for some scalar k
• Collinear (C on line AB): OC = a + t(b − a) for some scalar t
• Equal lengths: |XY| = |UV|
• Perpendicular: XY · UV = 0
Step 3: Compute and interpret the result in geometric terms.
Worked Example — Diagonals of a Parallelogram Bisect Each Other
Let ABCD be a parallelogram. Let OA = a, OB = b, OD = d. Since ABCD is a parallelogram, AB = DC.
AB = b − a. Since DC = AB: OC − OD = b − a, so OC = d + b − a.
Midpoint of diagonal AC: ½(a + OC) = ½(a + d + b − a) = ½(b + d)
Midpoint of diagonal BD: ½(b + d)
The two midpoints are equal, so the diagonals bisect each other. ✓
The Strategy for Vector Proofs
A vector proof has four stages:
1. Identify what to prove. Is it a midpoint relationship? Parallelism? Perpendicularity? Collinearity? Knowing this tells you what to compute.
2. Set up coordinates. Assign position vectors to the vertices. Choose an origin that simplifies the algebra. For a triangle ABC, placing O at A (so a = 0) is often elegant, but placing O outside the figure (as a separate origin) gives more symmetric expressions.
3. Express everything and compute. Write all required points and vectors in terms of your base position vectors. Then compute — addition, subtraction, dot products, magnitudes as needed.
4. Interpret the result. State clearly what the calculation proves in geometric terms.
Parallel Lines: Checking for Multiples
Two vectors are parallel iff one is a scalar multiple of the other. In a proof, you’ll often arrive at an expression like MN = ½(c − b). If BC = c − b, then you can immediately state that MN = ½BC, and hence MN is parallel to BC with half the length.
Collinear Points
To show three points P, Q, R are collinear, show that PQ = kPR (or PR = kPQ) for some scalar k. This says the vectors from P to Q and from P to R are parallel — and since they share the point P, all three points lie on the same line.
Using the Dot Product for Perpendicularity
To prove two lines are perpendicular, compute the dot product of their direction vectors and show it equals zero. This is used in many classical geometry results, including: angle in a semicircle = 90°; diagonals of a rhombus are perpendicular; altitude of an equilateral triangle bisects the base at right angles.
The Power of Abstract Proof
The great advantage of vector proofs is that they work for any configuration, not just specific coordinates. When you prove the midpoint theorem using vectors a, b, c, the proof holds for every triangle in the plane simultaneously. This is the essence of mathematical proof.
Mastery Practice
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Fluency
Q1 — Triangle Midpoint Theorem
Triangle ABC has OA = a, OB = b, OC = c. M is the midpoint of AB and N is the midpoint of AC. Prove that MN ∥ BC and that MN = ½BC.
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Fluency
Q2 — Diagonals of a Parallelogram
ABCD is a parallelogram with OA = a, OB = b, OD = d. (a) Find OC in terms of a, b, d. (b) Prove that the diagonals AC and BD bisect each other.
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Fluency
Q3 — Centroid of a Triangle
Triangle ABC has OA = a, OB = b, OC = c. The medians are the lines from each vertex to the midpoint of the opposite side. Show that all three medians pass through the point G with position vector OG = ⅓(a + b + c).
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Fluency
Q4 — Diagonals of a Rhombus are Perpendicular
ABCD is a rhombus (a parallelogram with equal side lengths). Let A be the origin, AB = b, AD = d, with |b| = |d| (equal sides). The diagonals are AC and BD. Prove that AC ⊥ BD.
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Understanding
Q5 — Angle in a Semicircle
O is the centre of a circle with radius r. A and B are opposite ends of a diameter, so OA = a and OB = −a. P is any point on the circle, so OP = p with |p| = |a| = r. Prove that angle APB = 90°.
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Understanding
Q6 — Length of a Median
In triangle OAB with OA = a and OB = b, M is the midpoint of AB. Find the length of the median OM in terms of |a|, |b|, and a · b.
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Understanding
Q7 — Prove a Quadrilateral is a Parallelogram
Points A = (1, 2), B = (4, 3), C = (5, 6), D = (2, 5). Use vectors to prove ABCD is a parallelogram.
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Understanding
Q8 — Point on a Line Segment
Show that P = (3, 4) lies on the line segment from A = (1, 2) to B = (7, 8), and find the ratio AP : PB.
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Problem Solving
Q9 — Parallelogram: Diagonals and Intersection
ABCD is a quadrilateral with A = (0, 0), B = (4, 0), C = (5, 3), D = (1, 3). (a) Show ABCD is a parallelogram using vectors. (b) Find the point where the diagonals intersect and verify it is the midpoint of both diagonals.
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Problem Solving
Q10 — Intersection of Medians
Triangle OAB has O at the origin, OA = a, OB = b. M is the midpoint of OB and the median from A goes to M. N is the midpoint of OA and the median from B goes to N. Find the intersection of the two medians AM and BN, and show it equals ⅓(a + b).