Position Vector of a point dividing a line segment in a given ratio Calculator
Calculates the position vector of a point that splits a line segment between two given points in a specified internal ratio.
Formula first
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.
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
Apply it well
When To Use
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.
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.
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.
Solve for:
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.
References
Sources
- A-Level Mathematics Pure Mathematics Year 2, Section: Vectors
- Stewart, J. (2015). Calculus: Early Transcendentals, 8th Edition
- Pearson Edexcel International A Level Mathematics Pure Mathematics Student Book 2