When P divides AB in the ratio AP : PB = 2 : 3, the vector from A to P is two fifths of the vector from A to B. Start from A, count two of the five equal parts, and add that movement to the position vector of A if the question asks for the position of P.
The 2027 SEC G3 Mathematics K310 syllabus includes vector arithmetic and geometry problems involving vectors. This guide shows how the same point can be reached from either endpoint and how to check which fraction goes with each direction.
- Add the ratio terms to find the number of parts in AB.
- Match the starting point to its share of the line.
- Subtract position vectors in the order destination minus start.
- Check the answer by reaching P from the other endpoint.
CHAPTER 1 OF 4 · Back to contents
Distinguish a position from a movement
The position vector of a point tells you how to get there from the chosen origin O. The displacement AB tells you how to get from A to B. If A has position vector a and B has position vector b, travelling from A to B adds b − a.
Read AP : PB aloud as ‘from A to P compared with from P to B’. When the ratio is 2 : 3 and P lies on the segment, AP takes two parts and PB takes three parts of the whole five. Both AP and PB point in the same direction as AB.
The vector BP points back towards A. Its length is three fifths of AB, so BP is three fifths of BA. Mark the arrow on your sketch before deciding the sign.
CHAPTER 2 OF 4 · Back to contents
Worked example | Reach the same point two ways
Let A be (1, 2) and B be (11, 7). P lies on AB with AP : PB = 2 : 3. The vector AB has components (10, 5). Two fifths of that movement has components (4, 2), so the coordinates of P are (1, 2) + (4, 2) = (5, 4).
| Start | Movement | Result |
|---|---|---|
| A at (1, 2) | ⅖ of AB = (4, 2) | P at (5, 4) |
| B at (11, 7) | ⅗ of BA = (−6, −3) | P at (5, 4) |
The movement from P to B is (6, 3). Both displacements are multiples of (2, 1). AP is twice that vector and PB is three times that vector, so the ratio checks without calculating either distance.
In an answer written with vectors, put an arrow over a pair of point labels or use the notation in your question. Use parentheses for the coordinates of the resulting point.
Contents · Previous chapter · Next chapter
CHAPTER 3 OF 4 · Back to contents
Practise when the point is beyond B
Keep A and B fixed. Q lies on AB produced beyond B, and AB : BQ = 5 : 2. Starting from B, move another two fifths of AB in the same direction. That adds (4, 2), giving Q = (15, 9).
Starting from A, the whole journey AQ has seven parts while AB has five. Taking seven fifths of AB gives (14, 7), which added to A also gives (15, 9). Read the wording to decide whether to add the ratio parts or whether the whole segment is already one of the named quantities.
Check that Q is beyond B on the sketch. A calculation that gives a point between A and B has used the wrong part of the line.
Contents · Previous chapter · Next chapter
CHAPTER 4 OF 4 · Back to contents
Frequently asked questions
Can I use three fifths from A in the first example?
Three fifths is the share from B to P. Starting from A calls for two fifths.
What if the line slopes downwards?
Use the signed components of the movement. A downward component stays negative when multiplied by a positive fraction.
What if I am asked for AP only?
Give the displacement from A to P. Add the position of A only when you need the position of P.
Contents · Previous chapter · Continue with the related guide