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MathidiaMathematics Academy
Vectors and geometric proof
Lesson 48 of 75
Standard practice

Dividing lines in a ratio

Objective. Locate internal division points using vectors.

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  2. 2. Worked example
  3. 3. Practice
  4. 4. Feedback
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Learn: the key idea

Dividing lines in a ratio uses vectors to locate a point that splits a line segment AB in a given ratio, such as m:n. The position vector of the dividing point P is OP = OA + (m/(m+n))(OB − OA), which moves the correct fraction of the way along AB from A towards B. If A = (0, 0), B = (10, 5) and P divides AB in the ratio 2:3, then OP = (0,0) + (2/5)(10,5) = (4, 2).

Worked example

Point A has position vector (2, 1) and point B has position vector (12, 11). Find the position vector of point P dividing AB in the ratio 3:2.

2 … 1
Illustrated card showing 2 … 1
  1. 1Find AB = OB − OA = (12−2, 11−1) = (10, 10)
  2. 2Find the fraction along AB for P: 3/(3+2) = 3/5
  3. 3Scale AB by this fraction: (3/5)(10, 10) = (6, 6)
  4. 4Add to OA: OP = (2, 1) + (6, 6) = (8, 7)

Answer: (8, 7)

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Common mistake. Students often use the wrong fraction, such as n/(m+n) instead of m/(m+n), when finding the point closer to A for ratio m:n. Always check which part of the ratio corresponds to the distance from A before choosing the fraction.

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