Position Vector of a point dividing a line segment in a given ratio Equation, Derivation & Rearrangements

Core idea

Overview

This formula allows for the geometric division of space by expressing a point P between A and B as a weighted average of their position vectors. The ratio λ:μ signifies that AP:PB = λ:μ, effectively distributing the weight of each endpoint based on their proximity to the target point. It is a fundamental tool for solving problems involving collinearity and geometry in three-dimensional space.

When to use: Use this when you need to find the coordinates or vector of a point lying on a line segment between two known points at a specific proportional distance.

Why it matters: It is essential for computer graphics in vertex interpolation, structural engineering for determining stress points on beams, and navigation systems calculating midpoints or partition points along a route.

Symbols

Variables

\mathbf{p} = Position vector of P, \mathbf{a} = Position vector of A, \mathbf{b} = Position vector of B, \lambda = Ratio part 1, \mu = Ratio part 2

p

Position vector of P

V a r iab l e

a

Position vector of A

V a r iab l e

b

Position vector of B

V a r iab l e

λ

Ratio part 1

V a r iab l e

μ

Ratio part 2

V a r iab l e

Walkthrough

Derivation

Derivation of Position Vector of a point dividing a line segment in a given ratio

This derivation uses vector addition and the geometric property of proportional segments to express the position vector of a point dividing a line segment in a specific ratio.

A and B are distinct points with position vectors a and b relative to an origin O.
Point P lies on the line segment AB such that the ratio AP:PB is λ:μ.

1

Define the ratio property

Translate the given ratio λ:μ into an algebraic relationship between the vectors AP and PB.

\frac{A P}{P B} = \frac{λ}{μ} ⟹ μ A P = λ P B

Note: Ensure the order of points is consistent with the ratio given.

2

Express vectors in terms of position vectors

Substitute the position vectors of the points P, A, and B (p, a, and b) into the vector expressions AP and PB.

μ (p - a) = λ (b - p)

Note: Recall that any vector XY can be written as y - x.

3

Expand and rearrange terms

Distribute the scalars μ and λ to isolate the position vector p.

μ p - μ a = λ b - λ p

Note: Be careful with signs during expansion.

4

Solve for p

Group all terms containing p on the left side, factor out p, and divide by the sum of the ratio components.

p (λ + μ) = μ a + λ b ⟹ p = \frac{μ a + λ b}{λ + μ}

Note: A common mistake is swapping λ and μ; remember that the ratio λ:μ is 'crossed' with the vectors b and a respectively.

Result

p (λ + μ) = μ a + λ b ⟹ p = \frac{μ a + λ b}{λ + μ}

Source: Pearson Edexcel International A Level Mathematics Pure Mathematics Student Book 2

Free formulas

Rearrangements

Solve for $p$

Make p the subject

p = \frac{μ a + λ b}{λ + μ}

This is the original form of the formula, already solving for the position vector p.

Difficulty: 1/5

Solve for $a$

Make a the subject

a = \frac{p ( λ + μ ) - λ b}{μ}

Rearrange the equation to solve for the position vector a of point A.

Difficulty: 3/5

Solve for $b$

Make b the subject

b = \frac{p ( λ + μ ) - μ a}{λ}

Rearrange the equation to solve for the position vector b of point B.

Difficulty: 3/5

Solve for $λ$

Make lambda the subject

λ = \frac{μ ( a - p )}{p - b}

Rearrange the equation to solve for the ratio component lambda.

Difficulty: 4/5

Solve for $μ$

Make mu the subject

μ = \frac{λ ( b - p )}{p - a}

Rearrange the equation to solve for the ratio component mu.

Difficulty: 4/5

The static page shows the finished rearrangements. The app keeps the full worked algebra walkthrough.

Why it behaves this way

Intuition

Think of this as a 'weighted average' of two locations. If you are standing at point A and want to reach point B, you are pulling the position towards B by a factor of λ and towards A by a factor of μ. The point P is the 'center of mass' of two particles placed at A and B with weights μ and λ, respectively, which naturally lands P at the specific division point on the line segment.

p

Position vector of point P

The 'target' coordinate we are trying to find relative to the origin.

a, b

Position vectors of endpoints A and B

The two fixed 'anchor' points in space that define the start and end of the segment.

λ, μ

Ratio components

The 'weights' assigned to each end; if you want to be closer to B, you increase λ relative to μ.

Signs and relationships

λ + μ: This acts as a normalization factor (the total 'mass'). Dividing by the sum ensures that the weights are relative and that if λ=μ (the midpoint), the result is exactly the average of the two vectors.
μ\mathbf{a} + λ\mathbf{b}: The cross-multiplication (λ paired with b and μ with a) is necessary because a higher weight on λ 'pulls' the point closer to b, effectively shifting the balance toward the B end of the segment.

One free problem

Practice Problem

Point A has position vector (2, 0) and point B has position vector (10, 8). Find the position vector of point P which divides AB in the ratio 1:3.

ax2

ay0

bx10

by8

Ratio part 11

Ratio part 23

Solve for: $p$

Hint: Apply the ratio λ=1 and μ=3 to the coordinates: P = (μa + λb) / (λ + μ).

The full worked solution stays in the interactive walkthrough.

Where it shows up

Real-World Context

A drone surveyor identifying a landmark exactly three-quarters of the way along a straight power line between two pylons at known GPS coordinates.

Study smarter

Tips

Ensure the ratio λ:μ corresponds to AP:PB, not AP:AB.
If the point is a midpoint, λ and μ are equal, simplifying the formula to (a+b)/2.
Check that your vector notation is consistent throughout the calculation.

Avoid these traps

Common Mistakes

Swapping the ratio values for the wrong vectors (multiplying λ by b instead of a).
Confusing the ratio AP:PB with the ratio of AP:AB.
Forgetting to divide by the sum (λ + μ) at the final step.

Keep going

Related Formulas

Common questions

Frequently Asked Questions

This derivation uses vector addition and the geometric property of proportional segments to express the position vector of a point dividing a line segment in a specific ratio.

Use this when you need to find the coordinates or vector of a point lying on a line segment between two known points at a specific proportional distance.

It is essential for computer graphics in vertex interpolation, structural engineering for determining stress points on beams, and navigation systems calculating midpoints or partition points along a route.

Swapping the ratio values for the wrong vectors (multiplying λ by b instead of a). Confusing the ratio AP:PB with the ratio of AP:AB. Forgetting to divide by the sum (λ + μ) at the final step.