Equation Of The Line That Is Perpendicular

The equation of a line that is perpendicular is a fundamental concept in coordinate geometry, with wide-ranging applications in fields like physics, engineering, computer graphics, and more. Understanding how to find and manipulate these equations is crucial for solving geometric problems and modeling real-world phenomena.

What Does Perpendicular Mean?

In geometry, perpendicularity describes the relationship between two lines that intersect at a right angle (90 degrees). Imagine the corner of a square or a perfectly crossed intersection – that's perpendicularity in action. When lines are perpendicular, their slopes have a specific relationship that we'll explore.

Slopes and Perpendicularity

The slope of a line, often denoted by m, measures its steepness and direction. It tells us how much the line rises (or falls) for every unit it moves horizontally. The slope is calculated as:

m = (change in y) / (change in x) = Δy / Δx

For two lines to be perpendicular, their slopes must be negative reciprocals of each other. This means that if one line has a slope of m1, the slope of a line perpendicular to it (m2) will be:

m2 = -1 / m1

In simpler terms, you flip the fraction and change the sign.

Example:

• If a line has a slope of 2/3, a perpendicular line will have a slope of -3/2.
• If a line has a slope of -5, a perpendicular line will have a slope of 1/5.
• If a line is horizontal (slope = 0), a perpendicular line will be vertical (undefined slope).

Forms of Linear Equations

Before diving into finding the equation of a perpendicular line, let's review the common forms of linear equations:

1. Slope-Intercept Form:

   • y = mx + b
   • Where m is the slope and b is the y-intercept (the point where the line crosses the y-axis).

2. Point-Slope Form:

   • y - y1 = m(x - x1)
   • Where m is the slope and (x1, y1) is a point on the line.

3. Standard Form:

   • Ax + By = C
   • Where A, B, and C are constants. While less directly useful for finding perpendicular slopes, it can be converted to slope-intercept form.

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

Here's a breakdown of how to find the equation of a line perpendicular to a given line:

Scenario 1: Given the Equation of a Line and a Point

Let's say you have a line with the equation y = 2x + 3 and you want to find the equation of a line perpendicular to it that passes through the point (1, 4).

Step 1: Determine the Slope of the Given Line

The given line is in slope-intercept form (y = mx + b), so the slope is easily identified. In this case, the slope of the given line (m1) is 2.

Step 2: Calculate the Slope of the Perpendicular Line

The slope of the perpendicular line (m2) is the negative reciprocal of m1.

m2 = -1 / m1 = -1 / 2

So, the slope of the perpendicular line is -1/2.

Step 3: Use the Point-Slope Form

You now have the slope of the perpendicular line (-1/2) and a point it passes through (1, 4). Plug these values into the point-slope form of a linear equation:

y - y1 = m(x - x1) y - 4 = (-1/2)(x - 1)

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

While the point-slope form is a valid equation, you might want to convert it to slope-intercept form (y = mx + b) for easier interpretation.

y - 4 = (-1/2)x + 1/2 y = (-1/2)x + 1/2 + 4 y = (-1/2)x + 9/2

Therefore, the equation of the line perpendicular to y = 2x + 3 and passing through the point (1, 4) is y = (-1/2)x + 9/2.

Scenario 2: Given Two Points on a Line and a Point for the Perpendicular Line

Suppose you have a line defined by the points (2, 5) and (4, 9), and you need to find the equation of a line perpendicular to it that passes through the point (-1, 2).

Step 1: Calculate the Slope of the Given Line

Use the slope formula with the two given points:

m1 = (y2 - y1) / (x2 - x1) = (9 - 5) / (4 - 2) = 4 / 2 = 2

Step 2: Calculate the Slope of the Perpendicular Line

As before, the slope of the perpendicular line (m2) is the negative reciprocal of m1.

m2 = -1 / m1 = -1 / 2

Step 3: Use the Point-Slope Form

You have the slope of the perpendicular line (-1/2) and a point it passes through (-1, 2). Plug these values into the point-slope form:

y - y1 = m(x - x1) y - 2 = (-1/2)(x - (-1)) y - 2 = (-1/2)(x + 1)

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

y - 2 = (-1/2)x - 1/2 y = (-1/2)x - 1/2 + 2 y = (-1/2)x + 3/2

Thus, the equation of the line perpendicular to the line passing through (2, 5) and (4, 9), and also passing through the point (-1, 2) is y = (-1/2)x + 3/2.

Scenario 3: Given the Standard Form of a Line and a Point

Let's say you're given a line in standard form: 3x + 4y = 7, and you want to find the equation of a line perpendicular to it that passes through the point (5, -2).

Step 1: Convert to Slope-Intercept Form

To easily determine the slope, convert the standard form to slope-intercept form (y = mx + b):

3x + 4y = 7 4y = -3x + 7 y = (-3/4)x + 7/4

Step 2: Determine the Slope of the Given Line

Now that the equation is in slope-intercept form, the slope (m1) is easily identified as -3/4.

Step 3: Calculate the Slope of the Perpendicular Line

The slope of the perpendicular line (m2) is the negative reciprocal of m1:

m2 = -1 / (-3/4) = 4/3

Step 4: Use the Point-Slope Form

You have the slope of the perpendicular line (4/3) and a point it passes through (5, -2). Plug these values into the point-slope form:

y - y1 = m(x - x1) y - (-2) = (4/3)(x - 5) y + 2 = (4/3)(x - 5)

Step 5: Simplify to Slope-Intercept Form (Optional)

y + 2 = (4/3)x - 20/3 y = (4/3)x - 20/3 - 2 y = (4/3)x - 26/3

Therefore, the equation of the line perpendicular to 3x + 4y = 7 and passing through the point (5, -2) is y = (4/3)x - 26/3.

Special Cases

• Horizontal Lines: A horizontal line has a slope of 0 (y = b). A line perpendicular to a horizontal line is a vertical line, which has an undefined slope and an equation of the form x = a (where a is a constant).

• Vertical Lines: A vertical line has an undefined slope (x = a). A line perpendicular to a vertical line is a horizontal line, which has a slope of 0 and an equation of the form y = b.

Proving Perpendicularity

If you are given the equations of two lines and want to prove they are perpendicular, follow these steps:

1. Determine the Slopes: Find the slopes (m1 and m2) of both lines. If the equations are in standard form, convert them to slope-intercept form first.

2. Check the Negative Reciprocal Relationship: Verify that m2 = -1 / m1. Alternatively, you can check if the product of the slopes is -1: m1 * m2 = -1.

If either of these conditions is true, the lines are perpendicular.

Common Mistakes to Avoid

• Forgetting the Negative Sign: It's easy to remember to flip the fraction but forget to change the sign. The perpendicular slope must be the negative reciprocal.

• Confusing Parallel and Perpendicular Slopes: Parallel lines have the same slope. Perpendicular lines have negative reciprocal slopes.

• Incorrectly Calculating Slope: Double-check your calculations when finding the slope using two points. Ensure you subtract the y-coordinates and x-coordinates in the correct order.

• Not Converting to Slope-Intercept Form: When given the standard form, forgetting to convert to slope-intercept form first can lead to errors in identifying the slope.

Applications of Perpendicular Lines

The concept of perpendicular lines is essential in numerous fields:

• Construction and Architecture: Ensuring walls are perpendicular to the floor and ceilings are parallel is crucial for structural integrity.

• Navigation: Calculating routes and bearings often involves perpendicular lines and angles.

• Computer Graphics: Perpendicular lines are used in rendering 3D objects, calculating lighting, and creating realistic shadows.

• Physics: Analyzing forces and motion frequently involves resolving vectors into perpendicular components.

• Engineering: Designing bridges, buildings, and other structures requires precise calculations involving perpendicular lines and angles.

Examples and Practice Problems

Example 1:

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

• Solution: The slope of the given line is -3. The perpendicular slope is 1/3. Using point-slope form: y - (-1) = (1/3)(x - 2). Simplifying to slope-intercept form: y = (1/3)x - 2/3 - 1 => y = (1/3)x - 5/3.

Example 2:

Determine if the lines 2x + 3y = 6 and 3x - 2y = 4 are perpendicular.

• Solution: Convert to slope-intercept form:
  • 2x + 3y = 6 => 3y = -2x + 6 => y = (-2/3)x + 2 (slope m1 = -2/3)
  • 3x - 2y = 4 => -2y = -3x + 4 => y = (3/2)x - 2 (slope m2 = 3/2)
  • Since (-2/3) * (3/2) = -1, the lines are perpendicular.

Practice Problems:

1. Find the equation of a line perpendicular to y = (5/2)x - 1 that passes through the point (-4, 3).
2. Find the equation of a line perpendicular to the line passing through points (1, 7) and (3, 1) that passes through the point (0, -5).
3. Are the lines x - 4y = 8 and 4x + y = 2 perpendicular? Why or why not?
4. A line is defined by the equation y = kx + 2. Find the value of k if this line is perpendicular to the line y = -4x + 1.
5. Find the equation of the perpendicular bisector of the line segment connecting the points (2, 4) and (6, 8). (Hint: You'll need to find the midpoint of the line segment first).

Advanced Concepts and Extensions

• Perpendicular Planes: The concept of perpendicularity extends to three-dimensional space. A line is perpendicular to a plane if it is perpendicular to every line in the plane that passes through its point of intersection with the plane.

• Orthogonal Vectors: In linear algebra, orthogonality is a generalization of perpendicularity to vectors. Two vectors are orthogonal if their dot product is zero.

• Perpendicular Curves: The concept of perpendicularity can also be applied to curves. Two curves are perpendicular at a point of intersection if their tangent lines at that point are perpendicular.

Conclusion

Understanding the equation of a perpendicular line is a cornerstone of geometry and a powerful tool in many related fields. By mastering the relationship between slopes, the different forms of linear equations, and the step-by-step process for finding perpendicular lines, you'll be well-equipped to tackle a wide range of geometric problems and real-world applications. Remember to practice regularly and pay attention to detail to avoid common mistakes. The more you work with these concepts, the more intuitive they will become.