If C lies beyond B on the line from A through B, the movement from A to C includes the whole movement from A to B plus the extra movement from B to C. Find the extra movement using the stated ratio. Then add it to the position vector of B.
The 2027 SEC G3 Mathematics K310 syllabus covers position vectors and geometric problems involving vectors. This example shows how a ratio changes when a point lies beyond the original segment.
- Draw A, B and C in order.
- Identify the two lengths in the ratio.
- Use arrows to check the direction of each movement.
- Add the movement to the correct starting position.
CHAPTER 1 OF 4 · Back to contents
Start with the part named in the question
Suppose AB : BC = 2 : 3. The segment from B to C is three halves as long as the segment from A to B. If both arrows point along the line from A towards C, the vector from B to C is 3/2 times the vector from A to B.
CHAPTER 2 OF 4 · Back to contents
Worked example | Find C when A and B are given
Take A = (1, 2) and B = (5, 4). The movement from A to B is (4, 2). Multiply by 3/2 to get the movement from B to C, (6, 3). Add this to B to obtain C = (11, 7).
Check from A. The movement from A to C is (10, 5), or 5/2 times the movement from A to B. The two pieces of the ratio add to five parts, and AB is two of those parts.
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CHAPTER 3 OF 4 · Back to contents
Read a different ratio on the same diagram
If the question gives AB : AC = 2 : 5 instead, multiply the vector from A to B by 5/2 to find the movement from A to C. Add it to A. This gives the same point because the ratio now describes the whole movement from A.
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CHAPTER 4 OF 4 · Back to contents
Frequently asked questions
When would the scalar be negative?
It is negative when the movement points opposite to the reference vector. For example, the movement from C back to B is −3/2 times the vector from A to B.
How can a parent check the method?
Ask the student to name where the arrow starts before adding it to a position vector.