Perpendicular Line Calculator

How to Find the Equation of a Perpendicular Line

In coordinate geometry, two lines in a Cartesian plane are perpendicular if they intersect at a right angle ($90^\circ$ or $\pi/2\text{ radians}$). The defining algebraic relationship between the slopes $m_1$ and $m_2$ of two non-vertical perpendicular lines is that their slopes are negative reciprocals of one another:

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

Given an initial line and a specific point $(x_0, y_0)$, you can uniquely determine the perpendicular line passing through that point.

Step-by-Step Calculation Process

1. Find the Slope of the Original Line ($m_1$)

Extract or compute the slope depending on how the original line is given:

• Slope-Intercept Form ($y = m_1 x + b_1$): The slope is the coefficient $m_1$.
• Standard Form ($Ax + By = C$): Solve for $y = -\frac{A}{B}x + \frac{C}{B}$, so $m_1 = -\frac{A}{B}$.
• Two Points $(x_1, y_1)$ and $(x_2, y_2)$: Compute $m_1 = \frac{y_2 - y_1}{x_2 - x_1}$.

2. Compute the Perpendicular Slope ($m_2$)

Take the negative reciprocal: $m_2 = -\frac{1}{m_1}$.

3. Special Cases: Horizontal and Vertical Lines

• Horizontal Lines ($y = c$, slope $m_1 = 0$): The perpendicular line is vertical with equation $x = x_0$ (slope undefined).
• Vertical Lines ($x = c$, slope $m_1 = \text{undefined}$): The perpendicular line is horizontal with equation $y = y_0$ (slope $m_2 = 0$).

4. Write the Equation in Various Forms

• Point-Slope Form: $$y - y_0 = m_2(x - x_0)$$
• Slope-Intercept Form: $$y = m_2 x + b_2, \quad \text{where } b_2 = y_0 - m_2 x_0$$
• Standard Form: $$-m_2 x + y = b_2 \implies A'x + B'y = C'$$

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