Hard

Ratio and Division of a Line Segment Using Vectors

Simple Explanation

This extends the midpoint idea: if a point P divides segment AB in the ratio m:n (measured from A to B), then P's position vector from any fixed origin O is a specific weighted combination of A and B's position vectors: OP = (n·OA + m·OB) / (m+n).

Why Do We Need It?

This lets you find the exact location of ANY point along a segment (not just the midpoint) purely with vector arithmetic — essential for later coordinate geometry and mechanics problems.

See It

Point P dividing segment AB in the ratio 2:3

Formula

The Ratio (Section) Formula

If P divides AB in the ratio m:n, then OP = (n·OA + m·OB) / (m+n)

Gives the position vector of a point P that divides the segment AB in a given ratio m:n, directly from the position vectors of A and B.

O: — the fixed origin
OA, OB, OP: — the position vectors of A, B, and P respectively
m : n: — the ratio in which P divides AB, measured from A to B

When to use it: Whenever a point divides a line segment in a known ratio and you need that point's position vector (e.g. the midpoint is the special case m=n=1).

Worked Example

Express a dividing point's position vector using the section formula

Point P divides segment AB such that AP:PB = 2:3. If OA and OB are the position vectors of A and B from origin O, express OP in terms of OA and OB.

Why Does This Work?

Starting from OP = OA + AP: since P divides AB in the ratio m:n, AP is the fraction m/(m+n) of the whole segment AB, so AP = (m/(m+n))·(OB−OA). Substituting and collecting terms gives OP = OA·(n/(m+n)) + OB·(m/(m+n)), which is exactly (n·OA + m·OB)/(m+n).

Real-Life Example

Animating smooth movement between two waypoints

A computer animation needs to place an object at a specific percentage of the way along a path between two fixed points.

This is exactly the ratio (section) formula in action — often called "lerp" (linear interpolation) by animators and game programmers.

Practice

Point P divides segment AB such that AP:PB = 1:4. If OP = (a·OA + b·OB) / 5, find the value of a.

Hard

Common mistake

Swapping m and n in the formula (using m·OA + n·OB instead of n·OA + m·OB) — the weight on each point comes from the ratio number of the OPPOSITE segment, not its own.

Quick Review

OP = (n·OA + m·OB) / (m+n), when AP:PB = m:n.
The midpoint is the special case m=n=1: OP = (OA+OB)/2.
The weight on each point is the ratio number from the OPPOSITE segment.