Vector Geometry: Proof and Ratio Problems
Vector geometry uses vectors to prove geometric facts, and it is one of the highest-scoring topics on the paper because the method is repeatable even when the diagram looks unfamiliar. Every journey around a shape can be written in terms of two base vectors, usually called a and b, by travelling along known edges: to get from X to Y, go backwards along the vector from the origin to X and forwards along the vector from the origin to Y. Two vectors are parallel when one is a scalar multiple of the other, and that single fact proves parallel lines. If two such vectors also share a point, the three points are collinear, meaning they lie on one straight line.
Before you start
Make sure you're comfortable with these topics first:
Method
- Label the base vectors clearly, for example the vector from O to A is a and the vector from O to B is b, and mark on the diagram every vector you work out.
- To find the vector from X to Y, use the route through a known point: the vector XY equals the vector from X back to the origin plus the vector from the origin to Y, that is minus x plus y.
- For a point dividing a line in a given ratio, travel to the start of the line and then along the fraction of it: if P divides AB in the ratio m to n, then OP = a + m/(m + n) times (b - a).
- For a midpoint, that formula reduces to a half of the sum of the two position vectors, which is worth remembering separately.
- To prove two lines are parallel, show one vector is a scalar multiple of the other, and say so explicitly: the vector XY equals k times the vector PQ, therefore XY is parallel to PQ.
- To prove three points are collinear, show that two vectors joining them are scalar multiples AND that they share a common point, then state both facts. Marks are lost for showing the multiple but not naming the shared point.
Worked example
In triangle OAB, the vector OA is a and the vector OB is b. M is the midpoint of AB, and P lies on OM such that OP is two thirds of OM. Find the vector OP in terms of a and b, and show that AP is parallel to the vector 2b - a.
- Find the vector AB: going from A back to O and then O to B gives -a + b, that is b - a.
- Find OM: M is the midpoint of AB, so OM = a + half of (b - a) = a + b/2 - a/2 = (a + b)/2.
- Find OP: P is two thirds of the way along OM, so OP = (2/3) times (a + b)/2 = (a + b)/3.
- Find AP: from A back to O and then O to P gives -a + (a + b)/3 = (-3a + a + b)/3 = (b - 2a)/3.
- Compare with 2b - a: the vector AP is (b - 2a)/3, which is not a multiple of 2b - a, so check the claim. In fact AP = (1/3)(b - 2a), so AP is parallel to b - 2a rather than to 2b - a.
- State the conclusion honestly: AP = (1/3)(b - 2a), so AP is parallel to the vector b - 2a, since one is a scalar multiple of the other. This is the kind of check worth doing before writing a proof, because a proof of a false statement cannot be rescued.
Practice questions
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Q1The vector OA is a and the vector OB is b. Write the vector AB in terms of a and b.Show answer
Answer: b - a.
Q2M is the midpoint of AB. Write OM in terms of a and b.Show answer
Answer: (a + b)/2, that is half of a plus half of b.
Q3How do you show two vectors are parallel?Show answer
Answer: Show that one is a scalar multiple of the other, then state that this means they are parallel.
Q4What extra fact is needed to prove three points are collinear rather than just that two vectors are parallel?Show answer
Answer: The two vectors must also share a common point, so the parallel segments lie on the same line rather than on two separate parallel lines.
Q5P divides AB in the ratio 1 to 3. Write OP in terms of a and b.Show answer
Answer: OP = a + (1/4)(b - a) = (3a + b)/4.
Q6The vector CD is 6a - 4b and the vector EF is 3a - 2b. Are CD and EF parallel?Show answer
Answer: Yes: CD = 2 times EF, so one is a scalar multiple of the other.
Q7Why is it useful to mark each vector you find on the diagram?Show answer
Answer: Later parts almost always build on earlier ones, and a marked diagram makes the next route obvious rather than something to re-derive.
Exam-style questions
Written in the style of a IGCSE Maths exam paper, with a full mark scheme.
OACB is a parallelogram with the vector OA = a and the vector OB = b. The point M is the midpoint of AC, and the point N lies on OC with ON one quarter of OC. Find the vector MN in terms of a and b, fully simplified.
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In triangle OAB, the vector OA = a and the vector OB = b. X is the point on OA with OX = (2/3)a, and Y is the point on OB with OY = (2/3)b. Prove that XY is parallel to AB.
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Free printable worksheet
Want more practice on paper? Download the vector geometry: proof and ratio problems worksheet pack - 7 pages of exam-style questions with a full mark scheme. One email opens every download in this browser for 14 days - no account, no card. Print it for personal and classroom use.
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