Condition of Perpendicularity

We will discuss here about the condition of perpendicularity of two straight lines.

Let the lines AB and CD be perpendicular to each other. If the inclination of AB with the positive direction of the x-axis is θ then the inclination of CD with the positive direction of the x-axis will be 90° + θ.

Therefore, the slope of AB = tan θ, and

the slope of CD = tan (90° + θ).

From trigonometry, we have, tan (90° + θ) = - cot θ

Therefore, if the slope of AB is m\(_{1}\) and

the slope CD = m\(_{2}\) then 

m\(_{1}\) = tan θ and m\(_{2}\) = - cot θ.

So, m\(_{1}\) ∙ m\(_{2}\) = tan θ ∙ (- cot θ) = -1

Two lines with slopes m\(_{1}\) and m\(_{2}\) are perpendicular to each other if and only if m\(_{1}\) ∙ m\(_{2}\) = -1

Note: (i) By the definition, the x-axis is perpendicular to the y-axis.

(ii) By definition, any line parallel to the x-axis is perpendicular to any line parallel to the y-axis.

(iii) If the slope of a line is m then any line perpendicular to it will have the slope \(\frac{-1}{m}\) (i.e., negative reciprocal of m).

 

Solved example on Condition of perpendicularity of two lines:

Find the equation of the line passing through the point (-2, 0) and perpendicular to the line 4x – 3y = 2.

Solution:

First we need to express the given equation in the form y = mx + c.

Given equation is 4x – 3y = 2.

-3y = -4x + 2

y = \(\frac{4}{3}\)x - \(\frac{2}{3}\)

Therefore, the slope (m) of the given line = \(\frac{4}{3}\)

Let the slope of the required line be m\(_{1}\).

According to the problem the required line is perpendicular to the given line.

Therefore, from the condition of perpendicularity we get,

m\(_{1}\) ∙ \(\frac{4}{3}\) = -1

⟹ m\(_{1}\) = -\(\frac{3}{4}\)

Thus, the required line has the slope -\(\frac{3}{4}\) and it passes through the point (-2, 0).

Therefore, using the point-slope form we get

y - 0 = -\(\frac{3}{4}\){x - (-2)}

⟹ y = -\(\frac{3}{4}\)(x + 2)

⟹ 4y = -3(x + 2)

⟹ 4y = -3x + 6

⟹ 3x + 4y + 6 = 0, which is the required equation.

 Equation of a Straight Line







10th Grade Math

From Condition of Perpendicularity of Two Straight Lines to HOME




Didn't find what you were looking for? Or want to know more information about Math Only Math. Use this Google Search to find what you need.



New! Comments

Have your say about what you just read! Leave me a comment in the box below. Ask a Question or Answer a Question.

Share this page: What’s this?

Recent Articles

  1. Intersecting Lines | What Are Intersecting Lines? | Definition

    Jun 14, 24 11:00 AM

    Intersecting Lines
    Two lines that cross each other at a particular point are called intersecting lines. The point where two lines cross is called the point of intersection. In the given figure AB and CD intersect each o…

    Read More

  2. Line-Segment, Ray and Line | Definition of in Line-segment | Symbol

    Jun 14, 24 10:41 AM

    Line-Segment, Ray and Line
    Definition of in Line-segment, ray and line geometry: A line segment is a fixed part of a line. It has two end points. It is named by the end points. In the figure given below end points are A and B…

    Read More

  3. Definition of Points, Lines and Shapes in Geometry | Types & Examples

    Jun 14, 24 09:45 AM

    How Many Points are There?
    Definition of points, lines and shapes in geometry: Point: A point is the fundamental element of geometry. If we put the tip of a pencil on a paper and press it lightly,

    Read More

  4. Subtracting Integers | Subtraction of Integers |Fundamental Operations

    Jun 13, 24 04:32 PM

    Subtraction of Integers
    Subtracting integers is the second operations on integers, among the four fundamental operations on integers. Change the sign of the integer to be subtracted and then add.

    Read More

  5. 6th Grade Worksheet on Whole Numbers |Answer|6th Grade Math Worksheets

    Jun 13, 24 04:17 PM

    6th Grade Worksheet on Whole Numbers
    In 6th Grade Worksheet on Whole Numbers contains various types of questions on whole numbers, successor and predecessor of a number, number line, addition of whole numbers, subtraction of whole number…

    Read More