How To Find Equation Of Perpendicular Line

Finding the equation of a perpendicular line is a fundamental concept in coordinate geometry, essential for various mathematical and real-world applications. Mastering this skill involves understanding slopes, reciprocals, and linear equations. This article will guide you through the process, providing clear steps and explanations.

Understanding Perpendicular Lines

Perpendicular lines are lines that intersect at a right angle (90 degrees). The key characteristic that distinguishes perpendicular lines from other intersecting lines lies in their slopes. The slope of a line describes its steepness and direction. For two lines to be perpendicular, the product of their slopes must be -1. This means that the slopes are negative reciprocals of each other.

Key Concepts

• Slope (m): The slope of a line is a measure of its steepness, calculated as the change in y divided by the change in x (rise over run). It is often represented by the variable m.
• Negative Reciprocal: The negative reciprocal of a number is found by flipping the fraction and changing its sign. For example, the negative reciprocal of 2 (or 2/1) is -1/2.
• Linear Equation: A linear equation is an equation that, when graphed, forms a straight line. The most common form is slope-intercept form: y = mx + b, where m is the slope and b is the y-intercept.

Steps to Find the Equation of a Perpendicular Line

Here’s a step-by-step guide to finding the equation of a line perpendicular to a given line:

Step 1: Determine the Slope of the Given Line

The first step is to identify the slope of the line you are given. This can be done in several ways, depending on the information provided:

• From the Equation (Slope-Intercept Form): If the equation of the line is in slope-intercept form (y = mx + b), the slope is simply the coefficient of x.

  • Example: If the equation is y = 3x + 5, the slope m is 3.

• From Two Points: If you are given two points on the line, ((x_1, y_1)) and ((x_2, y_2)), the slope can be calculated using the formula:

  [ m = \frac{y_2 - y_1}{x_2 - x_1} ]

  • Example: Given points (1, 2) and (4, 8), the slope m is:

    [ m = \frac{8 - 2}{4 - 1} = \frac{6}{3} = 2 ]

• From the Equation (Standard Form): If the equation is in standard form (Ax + By = C), you can rearrange it to slope-intercept form (y = (-A/B)x + C/B) to find the slope. The slope m is (-A/B).

  • Example: If the equation is 2x + 3y = 6, rearrange it to 3y = -2x + 6, and then y = (-2/3)x + 2. The slope m is -2/3.

Step 2: Calculate the Slope of the Perpendicular Line

Once you have the slope of the given line, calculate the negative reciprocal to find the slope of the perpendicular line.

• If the slope of the given line is m, the slope of the perpendicular line ((m_{\perp})) is:

  [ m_{\perp} = -\frac{1}{m} ]

  • Example 1: If the slope of the given line is 3, the slope of the perpendicular line is:

    [ m_{\perp} = -\frac{1}{3} ]

  • Example 2: If the slope of the given line is -2/3, the slope of the perpendicular line is:

    [ m_{\perp} = -\frac{1}{-\frac{2}{3}} = \frac{3}{2} ]

Step 3: Use the Point-Slope Form to Find the Equation

The point-slope form of a linear equation is:

[ y - y_1 = m(x - x_1) ]

where ((x_1, y_1)) is a point on the line and m is the slope.

• Substitute the slope of the perpendicular line ((m_{\perp})) and the coordinates of the given point into the point-slope form.

  • Example: Find the equation of a line perpendicular to y = 2x + 3 that passes through the point (4, 5).

    1. The slope of the given line is 2.

    2. The slope of the perpendicular line is (m_{\perp} = -\frac{1}{2}).

    3. Substitute (m_{\perp} = -\frac{1}{2}) and the point (4, 5) into the point-slope form:

       [ y - 5 = -\frac{1}{2}(x - 4) ]

• Simplify the equation to slope-intercept form (y = mx + b) or standard form (Ax + By = C), depending on the desired format.

  • Continuing from the previous example:

    [ y - 5 = -\frac{1}{2}x + 2 ]

    [ y = -\frac{1}{2}x + 7 ]

    The equation of the perpendicular line is y = -1/2x + 7.

Step 4: Verify the Solution

To ensure the equation is correct, you can verify the solution by:

• Checking the Slope: Ensure that the slope of the new line is indeed the negative reciprocal of the original line's slope.
• Substituting the Point: Substitute the coordinates of the given point into the new equation to ensure the point lies on the line.

Examples with Detailed Explanations

Let's work through several examples to solidify your understanding.

Example 1: Finding the Equation Given a Line and a Point

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

Solution:

1. Determine the slope of the given line:

   The slope of y = -3x + 4 is m = -3.

2. Calculate the slope of the perpendicular line:

   The slope of the perpendicular line is (m_{\perp} = -\frac{1}{-3} = \frac{1}{3}).

3. Use the point-slope form:

   Substitute (m_{\perp} = \frac{1}{3}) and the point (2, -1) into the point-slope form:

   [ y - (-1) = \frac{1}{3}(x - 2) ]

   [ y + 1 = \frac{1}{3}x - \frac{2}{3} ]

4. Simplify the equation to slope-intercept form:

   [ y = \frac{1}{3}x - \frac{2}{3} - 1 ]

   [ y = \frac{1}{3}x - \frac{5}{3} ]

   The equation of the perpendicular line is y = 1/3x - 5/3.

Example 2: Finding the Equation Given Two Points on the Original Line

Problem: Find the equation of a line perpendicular to the line passing through points (1, 5) and (3, -1) that passes through the point (0, 2).

Solution:

1. Determine the slope of the given line:

   Use the formula to find the slope of the line passing through (1, 5) and (3, -1):

   [ m = \frac{-1 - 5}{3 - 1} = \frac{-6}{2} = -3 ]

   The slope of the given line is m = -3.

2. Calculate the slope of the perpendicular line:

   The slope of the perpendicular line is (m_{\perp} = -\frac{1}{-3} = \frac{1}{3}).

3. Use the point-slope form:

   Substitute (m_{\perp} = \frac{1}{3}) and the point (0, 2) into the point-slope form:

   [ y - 2 = \frac{1}{3}(x - 0) ]

   [ y - 2 = \frac{1}{3}x ]

4. Simplify the equation to slope-intercept form:

   [ y = \frac{1}{3}x + 2 ]

   The equation of the perpendicular line is y = 1/3x + 2.

Example 3: Finding the Equation Given the Standard Form of the Original Line

Problem: Find the equation of a line perpendicular to 2x + 5y = 10 that passes through the point (-2, 3).

Solution:

1. Determine the slope of the given line:

   First, rearrange the equation to slope-intercept form:

   [ 5y = -2x + 10 ]

   [ y = -\frac{2}{5}x + 2 ]

   The slope of the given line is m = -2/5.

2. Calculate the slope of the perpendicular line:

   The slope of the perpendicular line is (m_{\perp} = -\frac{1}{-\frac{2}{5}} = \frac{5}{2}).

3. Use the point-slope form:

   Substitute (m_{\perp} = \frac{5}{2}) and the point (-2, 3) into the point-slope form:

   [ y - 3 = \frac{5}{2}(x - (-2)) ]

   [ y - 3 = \frac{5}{2}(x + 2) ]

4. Simplify the equation to slope-intercept form:

   [ y - 3 = \frac{5}{2}x + 5 ]

   [ y = \frac{5}{2}x + 8 ]

   The equation of the perpendicular line is y = 5/2x + 8.

Advanced Concepts and Special Cases

Horizontal and Vertical Lines

• Horizontal Lines: A horizontal line has a slope of 0. Its equation is in 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. Its equation is in the form x = c, where c is a constant. A line perpendicular to a vertical line is a horizontal line.

Example:

• Find the equation of a line perpendicular to y = 4 that passes through the point (3, 5).

  • The given line y = 4 is a horizontal line.
  • A line perpendicular to a horizontal line is a vertical line.
  • Since the perpendicular line must pass through the point (3, 5), its equation is x = 3.

Parallel and Perpendicular Lines

Understanding the relationship between parallel and perpendicular lines is crucial.

• Parallel Lines: Parallel lines have the same slope. If line 1 has a slope of m, a line parallel to it will also have a slope of m.
• Perpendicular Lines: As discussed, perpendicular lines have slopes that are negative reciprocals of each other.

Example:

• Given a line y = 2x + 1, find the equation of a line parallel to it that passes through the point (1, 4) and the equation of a line perpendicular to it that passes through the same point.

  • Parallel Line: The slope of the parallel line is the same as the given line, m = 2. Using the point-slope form:

    [ y - 4 = 2(x - 1) ]

    [ y = 2x + 2 ]

    The equation of the parallel line is y = 2x + 2.

  • Perpendicular Line: The slope of the perpendicular line is (m_{\perp} = -\frac{1}{2}). Using the point-slope form:

    [ y - 4 = -\frac{1}{2}(x - 1) ]

    [ y = -\frac{1}{2}x + \frac{9}{2} ]

    The equation of the perpendicular line is y = -1/2x + 9/2.

Applications of Perpendicular Lines

Perpendicular lines have various applications in mathematics, physics, engineering, and computer graphics.

• Geometry: Determining the shortest distance from a point to a line involves finding the perpendicular distance.
• Physics: Analyzing forces and motion often involves resolving vectors into perpendicular components.
• Engineering: Designing structures and ensuring stability often requires calculating perpendicular forces and distances.
• Computer Graphics: Creating realistic images and animations involves calculating angles and reflections using perpendicular lines.

Common Mistakes to Avoid

• Incorrectly Calculating the Negative Reciprocal: Make sure to both flip the fraction and change the sign when finding the negative reciprocal of the slope.
• Using the Original Slope Instead of the Perpendicular Slope: Always use the negative reciprocal slope when finding the equation of the perpendicular line.
• Algebra Errors: Double-check your algebra when simplifying equations to avoid mistakes in the final answer.
• Forgetting to Substitute the Point: Ensure you substitute the given point into the point-slope form correctly.

Practice Problems

To reinforce your understanding, here are some practice problems:

1. Find the equation of a line perpendicular to y = 4x - 3 that passes through the point (2, 1).
2. Find the equation of a line perpendicular to the line passing through points (-1, 2) and (3, 4) that passes through the point (0, -2).
3. Find the equation of a line perpendicular to 3x - 2y = 6 that passes through the point (1, -1).
4. Find the equation of a line perpendicular to y = -5 that passes through the point (-2, 3).
5. Find the equation of a line perpendicular to x = 2 that passes through the point (4, -1).

Conclusion

Finding the equation of a perpendicular line involves understanding the relationship between slopes, using the negative reciprocal, and applying the point-slope form of a linear equation. By following the steps outlined in this article and practicing with examples, you can master this essential concept in coordinate geometry. Remember to verify your solutions and avoid common mistakes to ensure accuracy. This skill is valuable in various fields and provides a solid foundation for more advanced mathematical concepts.