Writing the Equation of Two Perpendicular Lines: A Comprehensive Guide

In mathematics, the equation of two perpendicular lines is a fundamental concept in geometry and linear algebra. Understanding this concept is crucial for solving various mathematical and real-world problems.

Concept of Perpendicular Lines

The key property of perpendicular lines is that their slopes are negative reciprocals of each other. If the slope of one line is ( m ), the slope of a perpendicular line will be (-frac{1}{m}).

Slopes of Perpendicular Lines

Given a line with the equation ( y mx b ), where ( m ) is the slope and ( b ) is the y-intercept, the equation of a line perpendicular to it can be derived as:

Perpendicular Line Equation: ( y -frac{1}{m}x c )

Here, ( c ) represents the y-intercept of the second, perpendicular line.

Using Two-Point Form to Derive Perpendicular Line

The two-point form of a line is given by:

( y - y_1 frac{y_2 - y_1}{x_2 - x_1} (x - x_1) )

Here, the slope ( m ) is ( frac{y_2 - y_1}{x_2 - x_1} ). Using the fact that the slopes of two perpendicular lines are negative reciprocals, we can derive the equation of the perpendicular line.

Example Solutions

For an example, consider the following two lines:

Line 1: ( y mx b ) Line 2: ( y -frac{1}{m}x c )

For instance, if a line has the equation ( y mx ) with the y-intercept ( c ), the perpendicular line can be written as:

Line 1: ( y mx )
Line 2 (perpendicular): ( y -frac{1}{m}x )

General Form of Equations

The general form of two perpendicular lines can be expressed as:

Line 1: ( y ax b )
Line 2 (perpendicular): ( y -frac{1}{a}x c )

Special Cases

In special cases, the lines can be written as:

Example 1: ( x 0 ) and ( y 0 ) are the equations of two lines perpendicular to each other with the joint equation being ( xy 0 ). Example 2: If one line is ( x - y 0 ) with slope 1, a perpendicular line would have a slope of -1. The joint equation for this case is ( x^2 - y^2 0 ).

Conclusion

In summary, the equation of two perpendicular lines involves understanding the relationship between their slopes. By utilizing the concept of negative reciprocals, we can derive the equation of a perpendicular line given the equation of the original line.