Slope Of Perpendicular Lines - Derivation, Formula, Example

Slope of Perpendicular Lines

Slope of perpendicular lines are such that the slope of one line is the negative reciprocal of the slope of another line. If the slopes of the two perpendicular lines are m1, m2, then we can represent the relationship between the slope of perpendicular lines with the formula m1.m2 = -1. The product of the slope of perpendicular lines is -1.

Let us learn more about the slope of perpendicular lines, their derivation, with the help of examples, FAQs.

1.	What Is the Slope of Perpendicular Lines?
2.	Formula for Slope of Perpendicular Lines
3.	Derivation of Slope of Perpendicular Lines
4.	How to Find Slope of Perpendicular Lines?
5.	Examples on Slope of Perpendicular Lines
6.	Practice Questions
7.	FAQs on Slope of Perpendicular Lines

What Is the Slope of Perpendicular Lines?

Slope of a perpendicular line can be computed from the slope of a given line. The product of the slope of a given line and the slope of the perpendicular line is equal to -1. If the slope of a line is m1 and the slope of the perpendicular line is m2, then we have m1.m2 = -1.

The equations of two perpendicular lines are such that the coefficients of x and y are interchanged. For the equation of a line ax + by + c1 = 0, the equation of the perpendicular line is bx - ay + c2 = 0.

Formula for Slope of Perpendicular Lines

The formula for the slope of two perpendicular lines is that the product of the slopes of individual lines is equal to -1. If the slope of the individual lines is m1 and m2 respectively, then the formula to represent the slope of two perpendicular lines is m1.m2 = -1. 

Formula of Slope of Perpendicular Lines: m1.m2 = -1

Further, the slope of each of the perpendicular lines can be found from the equations of the lines, or from the points on the line. The slope of a line having slope-intercept form of the equation of a line - y = mx + c is m, and the slope of a line having a general equation of a line ax + by + c = 0 is -a/b. Also, the slope of the line passing through any two points (x1, y1), and (x2, y2) is m = (y2 - y1)/(x2 - x1).

Derivation of Slope of Perpendicular Lines

The slope of the perpendicular line can be derived from the formula of the angle between two lines. For two lines having slopes m1 and m2, the angle between the two lines is obtained using Tanθ.

Tanθ = (m1 - m2)/(1 + m1.m2)

The angle between two perpendicular lines is 90º, and we have Tan90º= ∞

Tan90º = (m1 - m2)/(1 + m1.m2)

∞ = (m1 - m2)/(1 + m1.m2)

n/20 = (m1 - m2)/(1 + m1.m2)

Here the denominator of the right hand side of the expression can be equalized to zero.

1 + m1.m2 = 0

m1.m2 = -1

m2 = -1/m1

Thus the slope of the perpendicular line is equal to the negative inverse of the slope of the given line.

How to Find Slope of Perpendicular Lines?

The slope of perpendicular lines can be calculated by knowing the slope of one of the two perpendicular lines. Here we take the equation of one of the perpendicular lines as the general form of the equation of a line. The general form of equation of a line is as follows.

ax + by + c = 0 

Let us convert this above equation into the slope-intercept form of the equation of a line. 

y = -ax/b - c/b

The slope of this line is m1 = -a/b, and we have the slope of perpendicular lines formula as m1.m2 = -1. Thus we can find the slope of the other perpendicular lines as follows.

(-a/b).m2 = -1

m2= b/a

Thus the required equation of slope of the perpendicular line is b/a.

Let us understand this with the help of a simple numeric example. The given equation of a line is 5x + 3y + 7 = 0. Now let us try to find the slope of the perpendicular line. 

Comparing this equation 5x + 3y + 7 = 0, with ax + by + c = 0, we have a = 5, b = 3. The slopes of perpendicular lines is m1 = -a/b = -5/3, and m2 = b/a = 3/5.

 

☛Related Topics

• Equation of a Line
• Negative Slope
• Positive Slope
• Slope Intercept Form
• Point Slope Form