How To Write A Perpendicular Line

Finding the perpendicular line to a given line is a fundamental skill in geometry and linear algebra, with applications ranging from architecture and engineering to computer graphics and video game design. This comprehensive guide will walk you through the steps, concepts, and underlying principles involved in writing the equation of a perpendicular line.

Understanding Perpendicular Lines

Perpendicular lines are lines that intersect at a right angle (90 degrees). The relationship between their slopes is crucial in determining the equation of a perpendicular line. If a line has a slope m, a line perpendicular to it has a slope of -1/m. This is known as the negative reciprocal. Understanding this concept is the cornerstone of writing equations for perpendicular lines.

Key Concepts:

• Slope: The measure of the steepness of a line, calculated as the change in y divided by the change in x (rise over run).
• Negative Reciprocal: The negative reciprocal of a number is found by inverting the number and changing its sign. For example, the negative reciprocal of 2 is -1/2, and the negative reciprocal of -3/4 is 4/3.
• Slope-Intercept Form: A linear equation written in the form y = mx + b, where m is the slope and b is the y-intercept.
• Point-Slope Form: A linear equation written in the form y - y1 = m(x - x1), where (x1, y1) is a point on the line and m is the slope.
• Standard Form: A linear equation written in the form Ax + By = C, where A, B, and C are constants, and A and B are not both zero.

Steps to Write the Equation of a Perpendicular Line

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

Step 1: Identify the Slope of the Given Line

The first step is to determine the slope of the original line. This can be done in several ways, depending on the information provided:

• If the equation is in slope-intercept form (y = mx + b): The slope is simply the coefficient of x, which is m.
• If the equation is in standard form (Ax + By = C): Convert it to slope-intercept form by solving for y. The slope will then be the coefficient of x. Specifically, m = -A/B.
• If you are given two points (x1, y1) and (x2, y2): Use the slope formula: m = (y2 - y1) / (x2 - x1).

Example 1a: Slope-Intercept Form

Given the line y = 3x + 5, the slope m is 3.

Example 1b: Standard Form

Given the line 2x + 3y = 6, convert it to slope-intercept form:

• 3y = -2x + 6
• y = (-2/3)x + 2

The slope m is -2/3.

Example 1c: Two Points

Given the points (1, 2) and (4, 8), calculate the slope:

• m = (8 - 2) / (4 - 1)
• m = 6 / 3
• m = 2

The slope m is 2.

Step 2: Calculate the Negative Reciprocal of the Slope

Once you have the slope of the original line, find its negative reciprocal. This will be the slope of the perpendicular line. Remember to invert the slope and change its sign.

• If the original slope is m, the perpendicular slope is -1/m.

Example 2a:

If the original slope is 3, the perpendicular slope is -1/3.

Example 2b:

If the original slope is -2/3, the perpendicular slope is 3/2.

Example 2c:

If the original slope is 2, the perpendicular slope is -1/2.

Step 3: Determine a Point on the Perpendicular Line

To write the equation of a line, you need a slope and a point. If the problem specifies a point that the perpendicular line must pass through, use that point. If not, you may need to choose a point or leave the equation in terms of a variable b (y-intercept) to represent all possible perpendicular lines.

Example 3a:

The perpendicular line must pass through the point (2, 4).

Example 3b:

The perpendicular line must pass through the point (-1, 3).

Step 4: Use Point-Slope Form or Slope-Intercept Form to Write the Equation

Now that you have the slope of the perpendicular line and a point it passes through, you can write the equation using either the point-slope form or the slope-intercept form.

• Point-Slope Form: y - y1 = m(x - x1)
• Slope-Intercept Form: y = mx + b

Using Point-Slope Form:

Plug the perpendicular slope and the given point into the point-slope form. Then, simplify the equation to obtain the slope-intercept form if desired.

Using Slope-Intercept Form:

Plug the perpendicular slope into the slope-intercept form y = mx + b. Then, substitute the coordinates of the given point into the equation and solve for b (the y-intercept). Finally, write the complete equation using the calculated m and b values.

Example 4a: Using Point-Slope Form

Original line: y = 3x + 5 (slope m = 3, perpendicular slope = -1/3)

Point: (2, 4)

• y - 4 = (-1/3)(x - 2) (Point-slope form)
• y - 4 = (-1/3)x + 2/3
• y = (-1/3)x + 2/3 + 4
• y = (-1/3)x + 14/3 (Slope-intercept form)

Example 4b: Using Slope-Intercept Form

Original line: 2x + 3y = 6 (slope m = -2/3, perpendicular slope = 3/2)

Point: (-1, 3)

• y = (3/2)x + b
• 3 = (3/2)(-1) + b
• 3 = -3/2 + b
• b = 3 + 3/2
• b = 9/2

The equation of the perpendicular line is y = (3/2)x + 9/2.

Example 4c: Using Point-Slope Form

Original line: Points (1, 2) and (4, 8) (slope m = 2, perpendicular slope = -1/2)

Point: (5, 1)

• y - 1 = (-1/2)(x - 5) (Point-slope form)
• y - 1 = (-1/2)x + 5/2
• y = (-1/2)x + 5/2 + 1
• y = (-1/2)x + 7/2 (Slope-intercept form)

Step 5: Convert to Standard Form (Optional)

If required, you can convert the equation from slope-intercept form to standard form (Ax + By = C) by rearranging the terms. Ensure that A, B, and C are integers, and A is non-negative.

Example 5a:

Slope-intercept form: y = (-1/3)x + 14/3

• Multiply by 3 to eliminate fractions: 3y = -x + 14
• Rearrange to standard form: x + 3y = 14

Example 5b:

Slope-intercept form: y = (3/2)x + 9/2

• Multiply by 2 to eliminate fractions: 2y = 3x + 9
• Rearrange to standard form: -3x + 2y = 9
• Multiply by -1 to make A non-negative: 3x - 2y = -9

Example 5c:

Slope-intercept form: y = (-1/2)x + 7/2

• Multiply by 2 to eliminate fractions: 2y = -x + 7
• Rearrange to standard form: x + 2y = 7

Special Cases

• Horizontal Lines: A horizontal line has a slope of 0 (y = constant). A line perpendicular to a horizontal line is a vertical line (x = constant).
• Vertical Lines: A vertical line has an undefined slope (x = constant). A line perpendicular to a vertical line is a horizontal line (y = constant).

Example: Horizontal Line

Given the line y = 4, which is a horizontal line. A line perpendicular to it will be a vertical line. If the perpendicular line passes through the point (2, 3), the equation of the perpendicular line is x = 2.

Example: Vertical Line

Given the line x = -1, which is a vertical line. A line perpendicular to it will be a horizontal line. If the perpendicular line passes through the point (5, -2), the equation of the perpendicular line is y = -2.

Practice Problems

Let's solidify your understanding with some practice problems:

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

Solution:

1. The slope of the given line is -2.
2. The perpendicular slope is 1/2.
3. Using point-slope form: y - (-1) = (1/2)(x - 1)
4. Simplifying: y + 1 = (1/2)x - 1/2
5. Slope-intercept form: y = (1/2)x - 3/2
6. Standard form: x - 2y = 3

Problem 2: Find the equation of a line perpendicular to 3x - 4y = 8 and passing through the point (0, 5).

Solution:

1. Convert to slope-intercept form: -4y = -3x + 8 => y = (3/4)x - 2
2. The slope of the given line is 3/4.
3. The perpendicular slope is -4/3.
4. Using slope-intercept form: y = (-4/3)x + b
5. Plug in the point (0, 5): 5 = (-4/3)(0) + b => b = 5
6. Slope-intercept form: y = (-4/3)x + 5
7. Standard form: 4x + 3y = 15

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

Solution:

1. Find the slope of the given line: m = (1 - (-3)) / (4 - 2) = 4 / 2 = 2
2. The perpendicular slope is -1/2.
3. Using point-slope form: y - 2 = (-1/2)(x - (-2))
4. Simplifying: y - 2 = (-1/2)x - 1
5. Slope-intercept form: y = (-1/2)x + 1
6. Standard form: x + 2y = 2

Common Mistakes to Avoid

• Forgetting the Negative Reciprocal: The most common mistake is forgetting to both invert and change the sign of the slope.
• Incorrectly Calculating Slope: Double-check your slope calculation, especially when using the slope formula with two points.
• Using the Wrong Point: Make sure you are using the point that the perpendicular line passes through, not a point on the original line.
• Algebra Errors: Be careful with algebraic manipulations when converting between forms of the equation.
• Ignoring Special Cases: Remember the rules for horizontal and vertical lines.

Applications of Perpendicular Lines

Understanding and calculating perpendicular lines is useful in various real-world applications:

• Architecture and Construction: Ensuring walls are perpendicular to the ground, designing structures with right angles for stability.
• Engineering: Designing roads and bridges, ensuring proper alignment and stability.
• Computer Graphics: Creating 3D models, rendering images, and developing video games.
• Navigation: Determining optimal routes, calculating distances and angles.
• Physics: Analyzing forces and motion, understanding projectile trajectories.

Conclusion

Mastering the art of writing equations for perpendicular lines is a valuable skill in mathematics and beyond. By understanding the relationship between slopes, applying the negative reciprocal concept, and following the step-by-step guide, you can confidently solve a wide range of problems involving perpendicular lines. Remember to practice regularly and pay attention to the details to avoid common mistakes. With consistent effort, you'll be able to tackle any perpendicular line challenge with ease.