Slopes of Two Perpendicular Lines
A detailed tutorial on how to prove that the slopes of two perpendicular lines are the negative reciprocal of each other.
Problem:
Prove that two perpendicular lines have slopes that are negative reciprocal of each other.
Solution to Problem:

The figure below shows two perpendicular lines L1 and L2.
.

Let the equation of line L1 be
y = m_{1} x + b_{1}

We now select two points A and B on line L1 whose x coordinates are 6 and 8 respectively and find their y coordinates.
A(6 , 6 m_{1} + b_{1})
B(7 , 7 m_{1} + b_{1})

We now find the components of vector AB.
vector(AB) = > 7  6 , (7 m_{1} + b_{1})  (6 m_{1} + b_{1}) >
= > 1 , m_{1}>

Let y = m_{2} x + b_{2} be the equation of line L2. We now select two points C and D on line L2 whose x coordinates are 4 and 5 respectively and find their y coordinates.
A(4 , 4 m_{2} + b_{2})
B(5 , 5 m_{2} + b_{2})
vector(CD) = > 5  4 , (5 m_{2} + b_{2})  (4 m_{2} + b_{2}) >
= > 1 , m_{2}>

If the two lines are perpendicular then the two vectors AB and CD are orthogonal and their scalar product must be zero. Hence
> 1 , m_{1}> . > 1 , m_{2}> = 0

Which gives
1 + m_{1} * m_{2} = 0

The above may be written as
m_{1} * m_{2} =  1
m_{1} =  1 / m_{2} , m_{1} is equal to the negative reciprocal of m_{2}
m_{2} =  1 / m_{1} , m_{2} is equal to the negative reciprocal of m_{1}

NOTE: Points A, B, C and D and their x coordinates can choosen anywhere on the two lines. As an exercise choose other values for the x coordinates of points A, B, C and D and redo the calculation above. You should be able to make the same conclusion concerning the slopes of two perpendicular lines.
More math problems with detailed solutions ,
Match Linear Equations to Graphs. Excellent interactive activity where linear equations are matched to graphs.
Tutorial on Equation of Line
and Line Problems
Equations of Line Through Two Points And Parallel and Perpendicular.
 
Step by Step Math Worksheets SolversNew !
Linear ProgrammingNew !
Online Step by Step Calculus Calculators and SolversNew !
Factor Quadratic Expressions  Step by Step CalculatorNew !
Step by Step Calculator to Find Domain of a Function New !
Free Trigonometry Questions with Answers

Interactive HTML5 Math Web Apps for Mobile LearningNew !

Free Online Graph Plotter for All Devices
Home Page 
HTML5 Math Applets for Mobile Learning 
Math Formulas for Mobile Learning 
Algebra Questions  Math Worksheets

Free Compass Math tests Practice
Free Practice for SAT, ACT Math tests

GRE practice

GMAT practice
Precalculus Tutorials 
Precalculus Questions and Problems

Precalculus Applets 
Equations, Systems and Inequalities

Online Calculators 
Graphing 
Trigonometry 
Trigonometry Worsheets

Geometry Tutorials 
Geometry Calculators 
Geometry Worksheets

Calculus Tutorials 
Calculus Questions 
Calculus Worksheets

Applied Math 
Antennas 
Math Software 
Elementary Statistics
High School Math 
Middle School Math 
Primary Math
Math Videos From Analyzemath
Author 
email
Updated: February 2015
Copyright © 2003  2016  All rights reserved