Worked example 1
Divide a line internally
A(1, 2) and B(10, 8). Point P divides AB internally in the ratio AP:PB = 2:1. Find P.
- Use p = .
- Calculate each coordinate separately.
- Check P lies two-thirds of the way from A to B.
Answer: P = (7, 6)
Higher Mathematics
Higher Maths Collinearity and internal division
Practise collinearity and internal division for Scottish Higher Mathematics with worked examples, clear methods and original interactive questions. Prove collinearity and find points that divide a line internally in a given ratio.
Before you start
- Subtract position vectors.
- Use scalar multiples.
- Work with ratios.
Explanation
Points are collinear when direction vectors between them are scalar multiples. State the common point and scalar relationship clearly.
For internal division, weight each endpoint by its distance from the required point; the section formula avoids coordinate-by-coordinate guesswork.
Method and rules
- A, B and C are collinear if AB = kAC for some scalar k.
- If AP:PB = m:n, then p = .
- Check that an internal point lies between the endpoints.
Worked examples
Worked example 2
Method check
What vector relationship proves three points are collinear?
- They must have the same or opposite direction.
So: Two direction vectors sharing a point are scalar multiples.
Watch out
- Reversing the ratio weights.
- Claiming collinearity without showing a scalar multiple.
- Using position vectors as though they were lengths.
Exam reminder
For collinearity and internal division, show the defining equation or formula before simplifying. Check that every final value satisfies the original restrictions, interval or context.
Related Higher Maths topics
Continue with the methods that connect most closely to this topic.