Equation of a Line Perpendicular to a Given Line

Understanding the equation of a line that is perpendicular to a given line is a fundamental concept in coordinate geometry. When two lines are perpendicular to each other, they intersect at a right angle (90 degrees). The slopes of two perpendicular lines in a plane are negative reciprocals of each other.

Slope of a Line

The slope of a line is a measure of its steepness and is denoted by the letter 'm'. For a line in the Cartesian plane with the equation:

$$ y = mx + b $$

'm' represents the slope, and 'b' represents the y-intercept.

Slope of Perpendicular Lines

If we have two lines, Line 1 with slope $m_1$ and Line 2 with slope $m_2$, and these lines are perpendicular, then:

$$ m_1 \cdot m_2 = -1 $$

This means that:

$$ m_2 = -\frac{1}{m_1} $$

Equation of a Perpendicular Line

Given a line with the equation:

$$ y = m_1x + b_1 $$

A line perpendicular to this will have the equation:

$$ y = m_2x + b_2 $$

where $m_2 = -\frac{1}{m_1}$.

Table of Differences and Important Points

Aspect	Given Line	Perpendicular Line
Slope (m)	$m_1$	$m_2 = -\frac{1}{m_1}$
Angle of Intersection	-	90 degrees
Relationship	-	Negative reciprocal slopes

Formulas

• Slope of a given line: $m_1$
• Slope of a perpendicular line: $m_2 = -\frac{1}{m_1}$
• Equation of a given line: $y = m_1x + b_1$
• Equation of a perpendicular line: $y = m_2x + b_2$

Examples

Example 1: Find the Equation of a Line Perpendicular to a Given Line

Given the line $y = 2x + 3$, find the equation of a line perpendicular to it that passes through the point (1, -2).

Solution:

1. Find the slope of the given line: $m_1 = 2$.
2. Calculate the slope of the perpendicular line: $m_2 = -\frac{1}{m_1} = -\frac{1}{2}$.
3. Use the point-slope form of the line equation with point (1, -2) and slope $m_2$: $$ y - y_1 = m_2(x - x_1) $$ $$ y - (-2) = -\frac{1}{2}(x - 1) $$ $$ y + 2 = -\frac{1}{2}x + \frac{1}{2} $$ $$ y = -\frac{1}{2}x - \frac{3}{2} $$

The equation of the line perpendicular to $y = 2x + 3$ and passing through (1, -2) is $y = -\frac{1}{2}x - \frac{3}{2}$.

Example 2: Perpendicular to a Vertical Line

Find the equation of a line perpendicular to the vertical line $x = 5$.

Solution:

A vertical line has an undefined slope. A line perpendicular to a vertical line is a horizontal line, which has a slope of 0.

The equation of a horizontal line is of the form $y = k$, where 'k' is the y-coordinate of any point on the line.

Since the slope is 0, the equation of the line perpendicular to $x = 5$ is simply $y = k$, where 'k' can be any real number.

Example 3: Perpendicular to a Horizontal Line

Find the equation of a line perpendicular to the horizontal line $y = -3$ that passes through the point (4, 2).

Solution:

A horizontal line has a slope of 0. A line perpendicular to a horizontal line is a vertical line, which has an undefined slope.

The equation of a vertical line is of the form $x = h$, where 'h' is the x-coordinate of any point on the line.

Since the line must pass through (4, 2), the equation of the line perpendicular to $y = -3$ is $x = 4$.

By understanding the relationship between the slopes of perpendicular lines and applying the point-slope form of the line equation, you can find the equation of a line perpendicular to any given line.