Equation Of A Line Perpendicular To A Line

The equation of a line perpendicular to another is a fundamental concept in coordinate geometry, crucial for various applications in mathematics, physics, engineering, and computer graphics. Understanding how to derive and manipulate these equations allows us to analyze and solve a myriad of problems involving angles, distances, and spatial relationships. This article delves into the methods for determining the equation of a line perpendicular to a given line, complete with examples and explanations.

Understanding Perpendicular Lines

Two lines are said to be perpendicular if they intersect at a right angle (90 degrees). In coordinate geometry, the relationship between the slopes of perpendicular lines is particularly important.

Slopes of Perpendicular Lines

If a line has a slope m, then any line perpendicular to it will have a slope of -1/m, provided m ≠ 0. This relationship is based on the fact that the product of the slopes of two perpendicular lines is -1. Mathematically, if m₁ and m₂ are the slopes of two perpendicular lines, then:

m₁ * m₂ = -1

This means that if you know the slope of one line, you can easily find the slope of a line perpendicular to it by taking the negative reciprocal.

Methods to Find the Equation of a Perpendicular Line

There are several methods to find the equation of a line perpendicular to a given line. The choice of method depends on the information available, such as the slope and a point on the line, or the equation of the original line.

1. Using Slope-Intercept Form

The slope-intercept form of a linear equation is given by:

y = mx + b

Where:

• y is the dependent variable
• x is the independent variable
• m is the slope of the line
• b is the y-intercept (the point where the line crosses the y-axis)

Steps:

1. Determine the Slope of the Given Line: If the equation is in slope-intercept form, the slope is simply the coefficient of x. If the equation is in another form (e.g., standard form), rearrange it to slope-intercept form to identify the slope.

2. Find the Slope of the Perpendicular Line: Take the negative reciprocal of the slope of the given line. If the original slope is m, the perpendicular slope m_perp is -1/m.

3. Use the Point-Slope Form: If you have a point (x₁, y₁) through which the perpendicular line passes, use the point-slope form to find the equation:

   y - y₁ = m_perp (x - x₁)

4. Convert to Slope-Intercept Form (Optional): If desired, rearrange the equation to the slope-intercept form y = m_perp x + b by solving for y.

Example:

Find the equation of a line perpendicular to y = 2x + 3 that passes through the point (1, 2).

1. Slope of the Given Line: The slope of the given line is m = 2.

2. Slope of the Perpendicular Line: The slope of the perpendicular line is m_perp = -1/2.

3. Use the Point-Slope Form: y - 2 = -1/2 (x - 1)

4. Convert to Slope-Intercept Form: y - 2 = -1/2 x + 1/2 y = -1/2 x + 1/2 + 2 y = -1/2 x + 5/2

Therefore, the equation of the line perpendicular to y = 2x + 3 and passing through (1, 2) is y = -1/2 x + 5/2.

2. Using Standard Form

The standard form of a linear equation is given by:

Ax + By = C

Where:

• A, B, and C are constants

Steps:

1. Determine the Slope of the Given Line: Convert the standard form to slope-intercept form to find the slope, or use the formula m = -A/B.

2. Find the Slope of the Perpendicular Line: Take the negative reciprocal of the slope of the given line.

3. Use the Point-Slope Form: If you have a point (x₁, y₁) through which the perpendicular line passes, use the point-slope form to find the equation.

4. Convert to Standard Form (Optional): If desired, rearrange the equation to the standard form Ax + By = C.

Example:

Find the equation of a line perpendicular to 3x + 4y = 7 that passes through the point (2, -1).

1. Slope of the Given Line: The slope of the given line is m = -3/4.

2. Slope of the Perpendicular Line: The slope of the perpendicular line is m_perp = 4/3.

3. Use the Point-Slope Form: y - (-1) = 4/3 (x - 2) y + 1 = 4/3 (x - 2)

4. Convert to Standard Form: y + 1 = 4/3 x - 8/3 3(y + 1) = 3(4/3 x - 8/3) 3y + 3 = 4x - 8 -4x + 3y = -11 4x - 3y = 11

Therefore, the equation of the line perpendicular to 3x + 4y = 7 and passing through (2, -1) is 4x - 3y = 11.

3. Using Two Points on the Given Line

If you are given two points on the original line, you can find the slope using the formula:

m = (y₂ - y₁) / (x₂ - x₁)

Where:

• (x₁, y₁) and (x₂, y₂) are the coordinates of the two points.

Steps:

1. Calculate the Slope of the Given Line: Use the two given points to calculate the slope m.

2. Find the Slope of the Perpendicular Line: Take the negative reciprocal of the slope m to find m_perp.

3. Use the Point-Slope Form: If you have another point (x₃, y₃) through which the perpendicular line passes, use the point-slope form y - y₃ = m_perp (x - x₃).

4. Convert to Slope-Intercept or Standard Form (Optional): Rearrange the equation to the desired form.

Example:

Find the equation of a line perpendicular to the line passing through (1, 3) and (4, 5), and passing through the point (0, -2).

1. Calculate the Slope of the Given Line: m = (5 - 3) / (4 - 1) = 2/3

2. Slope of the Perpendicular Line: m_perp = -3/2

3. Use the Point-Slope Form: y - (-2) = -3/2 (x - 0) y + 2 = -3/2 x

4. Convert to Slope-Intercept Form: y = -3/2 x - 2

Therefore, the equation of the line perpendicular to the line passing through (1, 3) and (4, 5), and passing through the point (0, -2) is y = -3/2 x - 2.

Special Cases

1. Horizontal Lines

A horizontal line has a slope of m = 0. The equation of a horizontal line is of the form y = c, where c is a constant. A line perpendicular to a horizontal line is a vertical line.

Vertical Lines

A vertical line has an undefined slope. The equation of a vertical line is of the form x = k, where k is a constant.

Example:

Find the equation of a line perpendicular to y = 4 that passes through the point (2, 3).

Since y = 4 is a horizontal line, a line perpendicular to it must be vertical. The equation of the vertical line passing through (2, 3) is x = 2.

2. Vertical Lines

As mentioned, a vertical line has an undefined slope and is represented by the equation x = k. A line perpendicular to a vertical line is a horizontal line.

Example:

Find the equation of a line perpendicular to x = -1 that passes through the point (5, -2).

Since x = -1 is a vertical line, a line perpendicular to it must be horizontal. The equation of the horizontal line passing through (5, -2) is y = -2.

Advanced Concepts and Applications

1. Distance from a Point to a Line

The concept of perpendicular lines is crucial in finding the shortest distance from a point to a line. The shortest distance is along the line that is perpendicular to the given line and passes through the given point.

Steps:

1. Find the Equation of the Perpendicular Line: Determine the equation of the line perpendicular to the given line that passes through the given point.

2. Find the Intersection Point: Solve the system of equations formed by the given line and the perpendicular line to find their intersection point.

3. Calculate the Distance: Use the distance formula to find the distance between the given point and the intersection point.

Example:

Find the distance from the point (1, 2) to the line y = -x + 3.

1. Find the Equation of the Perpendicular Line: The slope of the given line is -1. The slope of the perpendicular line is 1. The equation of the perpendicular line passing through (1, 2) is: y - 2 = 1(x - 1) y = x + 1

2. Find the Intersection Point: Solve the system of equations: y = -x + 3 y = x + 1 Setting the two equations equal: x + 1 = -x + 3 2x = 2 x = 1 Then, y = 1 + 1 = 2 The intersection point is (1, 2).

3. Calculate the Distance: The distance between (1, 2) and (1, 2) is 0.

In this case, the point lies on the line, so the distance is 0.

2. Geometric Constructions

Perpendicular lines are fundamental in geometric constructions. For example, constructing a perpendicular bisector of a line segment involves finding a line that is perpendicular to the segment and passes through its midpoint.

3. Optimization Problems

In optimization problems, finding the minimum or maximum distance often involves using perpendicular lines. For instance, finding the point on a curve that is closest to a given point involves finding the line perpendicular to the tangent of the curve at that point.

Common Mistakes to Avoid

1. Forgetting to Take the Negative Reciprocal: One of the most common mistakes is forgetting to take the negative reciprocal of the slope when finding the slope of the perpendicular line.

2. Incorrectly Calculating the Slope: Make sure to calculate the slope correctly, especially when given two points.

3. Confusing Slope-Intercept and Standard Forms: Understand the differences between these forms and how to convert between them.

4. Not Using the Point-Slope Form Correctly: Ensure that the point (x₁, y₁) is correctly substituted into the point-slope form.

5. Algebraic Errors: Be careful with algebraic manipulations when rearranging equations.

Conclusion

Understanding and applying the concepts related to the equation of a line perpendicular to another is essential in coordinate geometry and its applications. By mastering the methods outlined in this article, one can solve a wide range of problems involving lines, distances, and angles. Whether using slope-intercept form, standard form, or points on a line, the key is to correctly identify the slope and apply the negative reciprocal relationship. With careful practice and attention to detail, you can confidently tackle any problem involving perpendicular lines.