Equation Of Line That Is Perpendicular

Let's dive into the world of perpendicular lines and their equations, a fundamental concept in coordinate geometry with applications in various fields, from architecture to computer graphics. Understanding the equation of a line perpendicular to another is essential for solving geometric problems and building a solid foundation in mathematics.

Understanding the Basics: Slope and Linear Equations

Before tackling perpendicular lines, let's recap some key concepts. The equation of a line represents a relationship between x and y coordinates of all points lying on that line. A common form of linear equation is the slope-intercept form:

• y = mx + b

Where:

• y is the dependent variable (typically plotted on the vertical axis).
• x is the independent variable (typically plotted on the horizontal axis).
• m is the slope of the line, representing its steepness and direction.
• b is the y-intercept, the point where the line crosses the y-axis (where x = 0).

The slope (m) is arguably the most crucial element when dealing with perpendicular lines. It's calculated as the "rise over run," which is the change in y divided by the change in x between any two points on the line:

• m = (y₂ - y₁) / (x₂ - x₁)

A positive slope indicates an increasing line (going upwards from left to right), a negative slope indicates a decreasing line, a slope of zero represents a horizontal line, and an undefined slope represents a vertical line.

What Does Perpendicular Mean?

In geometry, two lines are perpendicular if they intersect at a right angle (90 degrees). Visualize two lines crossing each other, forming a perfect "L" shape. This 90-degree intersection has a crucial implication for the slopes of these lines.

The Key Relationship: Slopes of Perpendicular Lines

The core concept is this: Two non-vertical lines are perpendicular if and only if the product of their slopes is -1. Mathematically:

• m₁ * m₂ = -1

Where:

• m₁ is the slope of the first line.
• m₂ is the slope of the second line (the one perpendicular to the first).

This relationship means that the slope of a line perpendicular to another is the negative reciprocal of the original line's slope. To find the negative reciprocal, you flip the fraction and change its sign.

Example:

If a line has a slope of 2/3, the slope of a line perpendicular to it is -3/2. (2/3) * (-3/2) = -1.

Finding the Equation of a Perpendicular Line: Step-by-Step

Here's a structured approach to determining the equation of a line perpendicular to a given line that passes through a specific point:

Step 1: Determine the Slope of the Given Line

If the equation of the given line is in slope-intercept form (y = mx + b), simply identify the coefficient of x, which is the slope (m). If the equation is in a different form (e.g., standard form: Ax + By = C), rearrange it into slope-intercept form to isolate y. Alternatively, if you have two points on the line, use the slope formula: m = (y₂ - y₁) / (x₂ - x₁).

Step 2: Calculate the Slope of the Perpendicular Line

Find the negative reciprocal of the slope you found in Step 1. If the original slope is m, the perpendicular slope (m⊥) is:

• m⊥ = -1/m

Step 3: Use the Point-Slope Form

The point-slope form of a linear equation is particularly useful when you know the slope of a line and a point it passes through:

• y - y₁ = m(x - x₁)

Where:

• (x₁, y₁) is the given point that the perpendicular line passes through.
• m is the slope of the perpendicular line (calculated in Step 2).

Step 4: Simplify to Slope-Intercept Form (Optional)

While the point-slope form is a valid equation for the line, you may want to convert it to slope-intercept form (y = mx + b) for easier interpretation and comparison. To do this, simply distribute the slope and solve for y.

Example:

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

1. Slope of the given line: The slope of y = 3x + 2 is 3.

2. Slope of the perpendicular line: The negative reciprocal of 3 is -1/3. So, m⊥ = -1/3.

3. Point-slope form: Using the point (1, 4) and the slope -1/3, we get: y - 4 = (-1/3)(x - 1)

4. Slope-intercept form (optional): y - 4 = (-1/3)x + 1/3 y = (-1/3)x + 1/3 + 4 y = (-1/3)x + 13/3

Therefore, the equation of the line perpendicular to y = 3x + 2 and passing through (1, 4) is y = (-1/3)x + 13/3.

Special Cases: Horizontal and Vertical Lines

The rule about negative reciprocals needs a slight adjustment when dealing with horizontal and vertical lines.

• Horizontal Lines: A horizontal line has a slope of 0. Its equation is always in the form y = c, where c is a constant (the y-intercept). A line perpendicular to a horizontal line is a vertical line.

• Vertical Lines: A vertical line has an undefined slope. Its equation is always in the form x = c, where c is a constant (the x-intercept). A line perpendicular to a vertical line is a horizontal line.

Key Takeaway: A horizontal line and a vertical line are always perpendicular to each other. You can't directly apply the negative reciprocal rule because you can't divide by zero (which would happen when finding the reciprocal of an undefined slope). Instead, remember the geometric relationship: horizontal and vertical lines are, by definition, perpendicular.

Example:

Find the equation of a line perpendicular to the line x = 5 that passes through the point (2, 3).

Since x = 5 is a vertical line, a line perpendicular to it must be horizontal. A horizontal line passing through (2, 3) will have the equation y = 3.

Alternative Forms of Linear Equations

While slope-intercept form (y = mx + b) is convenient, other forms of linear equations exist and can be useful:

• Standard Form (Ax + By = C): A, B, and C are constants. While not as directly informative about the slope and y-intercept, it's useful for certain algebraic manipulations and is often the desired form for final answers. To find the slope from standard form, rearrange to slope-intercept form or use the formula m = -A/B.

• Point-Slope Form (y - y₁ = m(x - x₁)): As mentioned before, this form is excellent when you know a point on the line and its slope.

Understanding how to convert between these forms is a valuable skill. For example, converting from standard form to slope-intercept form allows you to easily identify the slope and y-intercept.

Applications of Perpendicular Lines

The concept of perpendicular lines is not just an abstract mathematical idea; it has numerous practical applications:

• Architecture and Construction: Ensuring walls are perpendicular to the floor, designing structures with right angles for stability and aesthetics.
• Navigation: Determining routes and bearings, calculating the shortest distance between two points (which is along a line perpendicular to a given line).
• Computer Graphics: Creating realistic images, defining shapes and objects, performing transformations (rotations, reflections).
• Physics: Analyzing forces acting at right angles, calculating components of vectors.
• Coordinate Geometry Problems: Finding the distance from a point to a line, determining the equation of an altitude in a triangle.
• Robotics: Path planning, obstacle avoidance (robots often need to move perpendicularly to objects).

Common Mistakes to Avoid

• Forgetting the Negative Sign: Remember that the slope of the perpendicular line is the negative reciprocal. Don't just flip the fraction; change its sign as well.
• Confusing Parallel and Perpendicular Slopes: Parallel lines have the same slope. Perpendicular lines have slopes that are negative reciprocals of each other.
• Incorrectly Applying the Point-Slope Form: Make sure you are using the coordinates of the given point (x₁, y₁) correctly in the formula.
• Ignoring Horizontal and Vertical Lines: Remember the special case: the line perpendicular to a horizontal line is vertical, and vice versa.
• Algebra Errors: Double-check your algebraic manipulations when rearranging equations and solving for y.
• Not Reading the Question Carefully: Always understand what the question is asking. Are you looking for the equation of a perpendicular line, a parallel line, or something else?
• Assuming All Equations Must Be in Slope-Intercept Form: While slope-intercept form is useful, point-slope and standard forms are also valid and sometimes more convenient.

Practice Problems

To solidify your understanding, try these practice problems:

1. Find the equation of a line perpendicular to y = -2x + 5 that passes through the point (3, -1).

2. Find the equation of a line perpendicular to the line passing through the points (1, 2) and (4, 8) that passes through the point (-2, 5).

3. Find the equation of a line perpendicular to x = -3 that passes through the point (0, 0).

4. The line l has the equation 2x + 3y = 6. Find the equation of the line perpendicular to l that passes through the point (1, 1).

Answers:

1. y = (1/2)x - (5/2)

2. y = (-1/2)x + 4

3. y = 0

4. y = (3/2)x - (1/2)

Advanced Concepts and Extensions

• Distance from a Point to a Line: Finding the shortest distance from a point to a line involves finding the equation of the perpendicular line from the point to the line, finding the point of intersection, and then calculating the distance between the two points.
• Orthogonal Trajectories: In calculus, orthogonal trajectories are families of curves that intersect a given family of curves at right angles. Finding orthogonal trajectories involves differential equations and the concept of perpendicular slopes.
• Perpendicular Vectors: The concept of perpendicularity extends to vectors. Two vectors are perpendicular (or orthogonal) if their dot product is zero.
• Planes in 3D Space: In three-dimensional space, a line can be perpendicular to a plane. The direction vector of the line is parallel to the normal vector of the plane.

Conclusion

Understanding the equation of a line perpendicular to another line is a fundamental concept in coordinate geometry with broad applications. Mastering the relationship between slopes, using the point-slope form, and recognizing special cases involving horizontal and vertical lines are crucial skills. By practicing and applying these concepts, you can confidently solve a wide range of geometric problems and build a stronger foundation in mathematics. Remember the key: perpendicular lines have slopes that are negative reciprocals of each other, ensuring they intersect at a right angle.