Divide a line segment using vectors
Represent an interior point as a weighted combination of endpoint position vectors and use ratios to describe vector geometry.
Cambridge IGCSE Additional Mathematics 0606, 2028–2030 syllabus, Version 1. Paper 1 is non-calculator; Paper 2 requires a scientific calculator. These original lessons provide self-checked topic practice, not a Cambridge grade or a timed paper. Full-paper assessment and human specialist review are still outstanding.
Before you begin
- Cambridge assumes knowledge of IGCSE Mathematics or an equivalent course. Check the domain before using an inverse or logarithm, and retain exact values where the question asks for them.
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
If AP:PB = 3:1, how far along AB is P from A as a fraction of AB?
What you will learn
- Find a point's position vector when it divides a segment in a given ratio
- Use a midpoint as the equal-ratio case
- Check that the weighted position lies between the endpoint vectors
A point on a segment is a fraction of the way from one endpoint
If OA = a and OB = b, and P lies on AB with AP:PB = m:n, then OP = (n a + m b)/(m + n). The weight on a is the distance from P to B; the weight on b is the distance from A to P.
For AP:PB = 2:1, P is two-thirds of the way from A to B, so OP = (a + 2b)/3. The weights add to one, keeping P on the same segment.
The midpoint is the half-and-half case
If AP = PB, the ratio is 1:1 and OP = (a + b)/2. This agrees with averaging the endpoint coordinates.
Keep the stated ratio order attached to the points. AP:PB = 2:1 is not the same as 1:2; reversing the ratio changes which endpoint P is closer to.
Pause and explain
If AP:PB = 1:2, which endpoint is P closer to?
Worked example
OA = a and OB = b. Point P divides AB so that AP:PB = 2:1. Find OP.
Show the worked solution
- The ratio totals 3 parts; P lies two of those parts from A and one from B.
- Use OP = (1a + 2b)/3.
- The weights 1/3 and 2/3 sum to one, as expected for a point on AB.
Answer OP = (a + 2b)/3.
Try a topic test
Choose Easy, Normal or Difficult. Save your answers and open worked review after finishing.
Test this topic →From guided practice to a new situation
Write your answer and reasoning first. Use a hint only if you are stuck. The model solution is for self-checking; your written work is not automatically marked.
OA = a and OB = b. Point R divides AB with AR:RB = 3:1. Find OR.
Give me a hint
The ratio has four parts; R is three-quarters of the way from A to B.
Compare my reasoning
- Use OR = (1a + 3b)/(3 + 1).
- Simplify the denominator.
- Check that the weights add to one.
OR = (a + 3b)/4.
Look for these in your work
- Ratio order read correctly
- Weights assigned to endpoints
- Position remains on AB
M is the midpoint of A(−2, 5) and B(6, 1). Find M's coordinates and position vector.
Give me a hint
Average the corresponding endpoint coordinates.
Compare my reasoning
- The x-coordinate is (−2 + 6)/2 = 2.
- The y-coordinate is (5 + 1)/2 = 3.
- From the origin, the position vector is (2, 3).
M = (2, 3); position vector OM = (2, 3).
Look for these in your work
- Midpoint rule applied
- Coordinates averaged separately
- Position vector stated
Position vectors of P and R are p and r. Point Q lies on PR with PQ:QR = 1:2. Find OQ.
Give me a hint
Q lies one-third of the way from P to R.
Compare my reasoning
- The ratio totals 3; the weight on p is 2/3 and the weight on r is 1/3.
- Thus OQ = (2p + r)/3.
- The weights sum to one, placing Q between P and R.
OQ = (2p + r)/3.
Look for these in your work
- Endpoint weights reversed correctly
- Ratio converted to segment fraction
- Interior point check passed
Common mistakes
- Using the ratio weights in the wrong order
- Adding endpoint vectors without dividing by the total ratio
- Using an external-division formula for an interior point
- Forgetting that a midpoint gives equal weights
What can you explain now?
What is the position vector of the midpoint of AB if OA = a and OB = b?
Compare with the explanation
(a + b)/2
The midpoint is equally weighted from the two endpoints.
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Make it stick
How does a segment ratio determine the weights on its endpoint position vectors?
If you needed the solution, close it and solve the problem again from a blank page. A correct answer is stronger when you can explain the reason for each step.