Find the Equation of a Line Perpendicular to a Given Line that Passes Through a Given Point

Introduction

Key Concepts

Understanding Perpendicular Lines

Perpendicular lines are lines that intersect at a right angle (90 degrees). In the context of coordinate geometry, two lines are perpendicular if the product of their slopes is $-1$. This relationship is pivotal when determining the equation of a line perpendicular to a given line.

Slope of a Line

The slope of a line, often denoted as $m$, measures its steepness and direction. Given two points $(x_1, y_1)$ and $(x_2, y_2)$ on a line, the slope is calculated using the formula: $$ m = \frac{y_2 - y_1}{x_2 - x_1} $$ A positive slope indicates that the line ascends from left to right, while a negative slope indicates a descent.

Finding the Perpendicular Slope

If a line has a slope of $m$, any line perpendicular to it will have a slope of $-\frac{1}{m}$. This reciprocal relationship ensures that the two lines intersect at a right angle. For example, if the given line has a slope of $2$, the perpendicular slope will be $-\frac{1}{2}$.

Point-Slope Form of a Line

The point-slope form is a linear equation that uses a known point $(x_1, y_1)$ and the slope $m$ to define a line: $$ y - y_1 = m(x - x_1) $$ This form is particularly useful when you know a point through which the line passes and its slope.

Deriving the Equation of the Perpendicular Line

To find the equation of a line perpendicular to a given line and passing through a specific point, follow these steps:

1. Determine the slope of the given line.
2. Calculate the negative reciprocal of this slope to find the perpendicular slope.
3. Use the point-slope form with the perpendicular slope and the given point to write the equation.

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

1. The slope of the given line $y = 2x + 3$ is $m = 2$.
2. The perpendicular slope is $m' = -\frac{1}{2}$.
3. Using the point-slope form: $$ y - 5 = -\frac{1}{2}(x - 4) $$ Simplifying: $$ y = -\frac{1}{2}x + 7 $$

Special Cases

Certain scenarios require special attention:

• Vertical and Horizontal Lines: A vertical line has an undefined slope, and any line perpendicular to it must be horizontal with a slope of $0$. Conversely, a horizontal line has a slope of $0$, and any perpendicular line must be vertical.
• Given Point on the Line: Ensure that the given point lies on the newly derived perpendicular line by substituting its coordinates into the equation.

Graphical Interpretation

Graphing both the original and perpendicular lines can visually confirm their perpendicularity. The angle between them should be $90^\circ$. This visualization aids in reinforcing the theoretical understanding of perpendicular lines.

Applications in Real-Life Problems

This concept is widely applicable in various fields such as engineering, architecture, and computer graphics. For instance, designing perpendicular supports or understanding vector projections often requires determining perpendicular lines.

Advanced Concepts

Theoretical Foundations

The concept of perpendicular lines is deeply rooted in Euclidean geometry. The perpendicularity condition, where the product of slopes equals $-1$, arises from the geometric definition of orthogonality in the plane. This relationship ensures that the direction vectors of the two lines are orthogonal, a fundamental principle in vector geometry.

Mathematically, if two non-vertical lines are represented by $y = m_1x + c_1$ and $y = m_2x + c_2$, they are perpendicular if: $$ m_1 \cdot m_2 = -1 $$ This condition is derived from the requirement that the tangent of the angle between the two lines equals $-1$, indicating a $90^\circ$ angle.

Derivation of Perpendicular Slopes

Starting with the slope condition: $$ m_1 \cdot m_2 = -1 $$ Solving for $m_2$ gives: $$ m_2 = -\frac{1}{m_1} $$ This reciprocal relationship ensures that the lines are orthogonal. This derivation is foundational in linear algebra and plays a crucial role in defining orthogonal transformations.

Complex Problem-Solving

Consider a scenario where multiple lines and points are involved, requiring the determination of several perpendicular lines. For example:

1. Given two lines, find the equations of lines perpendicular to both that pass through two respective points.
2. Determine the perpendicular bisector of a line segment, which is crucial in constructing geometric shapes like triangles and polygons.

These problems necessitate a comprehensive understanding of slope relationships and adept application of the point-slope form.

Example: Find the equations of lines perpendicular to both $y = 3x - 2$ and $y = -\frac{1}{3}x + 4$ that pass through the point $(2, 3)$.

First, identify the slopes:

• Line 1: $m_1 = 3$
• Line 2: $m_2 = -\frac{1}{3}$

For a line to be perpendicular to both, it must satisfy: $$ m \cdot 3 = -1 \quad \text{and} \quad m \cdot \left(-\frac{1}{3}\right) = -1 $$ Solving these: $$ m = -\frac{1}{3} \quad \text{and} \quad m = 3 $$ However, these slopes correspond to the original lines, indicating no single line can be perpendicular to both unless they are parallel, which they are not. Therefore, no such line exists in this scenario.

Interdisciplinary Connections

The principles governing perpendicular lines extend beyond pure mathematics into fields like physics and engineering. In physics, vectors often require orthogonal components for analysis in multiple dimensions. In engineering, designing structures with perpendicular supports ensures stability and balance.

Moreover, in computer graphics, perpendicular lines are fundamental in rendering perspectives and shadows, contributing to realistic visual representations. Understanding these connections highlights the versatility and importance of perpendicularity across various disciplines.

Vector Representation

In vector algebra, lines can be represented using vectors, and perpendicularity is defined through the dot product. Two vectors $\vec{a}$ and $\vec{b}$ are perpendicular if: $$ \vec{a} \cdot \vec{b} = 0 $$ This condition translates to the slopes of their corresponding lines satisfying the perpendicular slope condition.

For example, consider vectors $\vec{a} = \langle a_1, a_2 \rangle$ and $\vec{b} = \langle b_1, b_2 \rangle$. Their dot product is: $$ \vec{a} \cdot \vec{b} = a_1b_1 + a_2b_2 $$ Setting this equal to zero ensures orthogonality.

Analytical Geometry Applications

Analytical geometry leverages the concept of perpendicular lines in various applications, such as finding the shortest distance between points and lines, optimizing paths, and solving geometric constructions. For instance, the shortest distance from a point to a line is found along the perpendicular from the point to the line.

This application is essential in areas like navigation, robotics path planning, and even in computer algorithms that require optimal pathfinding.

Comparison Table

Summary and Key Takeaways