Position vectors and division of a segment
Find directed segments from position vectors and locate points dividing a line in a stated ratio.
Extended only. This is the separate 9–1 qualification 0980. Cambridge intends Core for grades 4–1 and Extended for grades 9–5; the eligible award bands are grades 5–1 for the Core papers and grades 9–3 for the Extended papers.
Before you begin
- Add and subtract vectors.
- Interpret part-to-part ratios.
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
How do you express AB from positions a and b?
What you will learn
- Use AB = b − a from a common origin.
- Express an internal division point using two position vectors.
Subtract start position from end position
With OA = a and OB = b, the route A to O to B gives AB = −a + b. For a = (1,2), b = (7,5), AB = (6,3). Position vectors are measured from the same origin, while a segment vector begins at its named starting point.
The midpoint M has OM = a + ½(b − a) = ½(a + b). In the example it is (4,3.5). This is an average of positions; the vector AM is half the directed displacement AB, not half the absolute position of B.
Translate a ratio into a fraction of the whole
If AP:PB = 2:1, AP is 2/3 of AB. Therefore OP = a + 2/3(b − a) = 1/3 a + 2/3 b. For A(1,2), B(7,5), the point is (5,4). The ratio is not a factor of two for the entire AB segment.
Read the endpoint order carefully: reversing the ratio to 1:2 locates a different point. For an internal division, the coefficients of a and b sum to one and are both positive. On a diagram, check that the point lies between the endpoints and closer to the endpoint associated with the shorter segment.
Pause and explain
AP:PB = 2:1. What fraction of AB is AP?
Worked example
OA = a, OB = b and AP:PB = 2:1. Express OP.
Show the worked solution
- AP is two thirds of AB.
- AB = b − a, so OP = a + 2/3(b − a).
- Simplify to OP = 1/3 a + 2/3 b.
Answer OP = 1/3 a + 2/3 b.
Try a topic test
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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.
A(−1,3), B(5,−1). Find AB.
Give me a hint
Subtract A's coordinates from B's.
Compare my reasoning
- Horizontal displacement is 5 − (−1) = 6.
- Vertical displacement is −1 − 3 = −4.
- AB = (6,−4).
(6,−4).
Look for these in your work
- Used end minus start.
- Preserved direction.
A(1,2), B(7,5). Find P with AP:PB = 2:1.
Give me a hint
Start at A and move two thirds of AB.
Compare my reasoning
- AB = (6,3).
- Two thirds is (4,2).
- P = (1,2) + (4,2) = (5,4).
P(5,4).
Look for these in your work
- Converted the ratio correctly.
- Added the partial displacement to A.
OA = a, OB = b. M is the midpoint and N divides AB in ratio AN:NB = 1:3. Express MN.
Give me a hint
Find positions first, then subtract OM from ON.
Compare my reasoning
- OM = (a+b)/2; ON = a + (b−a)/4 = 3a/4+b/4.
- MN = ON − OM = a/4 − b/4.
- Thus MN = −(b−a)/4, pointing back toward A.
MN = (a − b)/4.
Look for these in your work
- Found both positions from one origin.
- Checked the backward direction.
Common mistakes
- Reversing end minus start.
- Using a part ratio as a whole fraction.
- Forgetting to add the starting position.
What can you explain now?
Why must position vectors share an origin?
Compare with the explanation
Their subtraction then represents the actual displacement.
Mixing origins adds an unaccounted shift and cannot directly produce AB.
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Make it stick
Tomorrow, derive an internal ratio-point position and check it using simple coordinates.
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.