Dividing a segment in a vector ratio
If AP:PB = m:n then position vector of P is (n a + m b)/(m+n).
Key formulas
Weights
P closer to A means n is larger in the numerator’s weight on a. Check with m:n=1:1.
Example
a=(0,0), b=(9,6), m:n=1:2 → p=(2·0+1·(9,6))/3=(3,2).
Proof sketch
From A to B, P = a + (m/(m+n))(b−a), which rearranges to the same formula.
Worked example
Worked example
Problem. A = (0,0), B = (9,6). Point P divides AB in the ratio 1 : 2. Find P.
Solution. p = (2·A + 1·B)/3 = (9,6)/3 = (3,2).
Diagram
Mark m parts toward B and n parts toward A on a number line first.
Review quiz
A=(4,5), B=(14,20), ratio 3:2. Position of P is:
Practice questions
- a=(2,2), b=(8,5), ratio 2:1. Find p.
- Show the formula gives the midpoint when m=n.
Study notes
External division?
Use (n a − m b)/(n−m) with care when n≠m.