Parallel and Perpendicular Line Calculator
Compute the equation of a line perpendicular to a given line through a point; handles slopes, vertical and horizontal cases.
Inputs
Actions
Equation Preview
Helping Notes
Notes: Perpendicular slope = negative reciprocal of original slope (m_perp = -1/m). Vertical and horizontal lines are handled specially.
If the original line is vertical (x = c), the perpendicular line is horizontal (y = y₀). If the original line is horizontal (y = c), result is vertical (x = x₀).
Results
Resulting slope:
—Point-slope form:
Slope-intercept form:
General form (Ax + By + C = 0):
Steps / Notes:
What is Parallel and Perpendicular Line Calculator
A Parallel and Perpendicular Line Calculator is a focused math tool that determines the equation and slope of a line perpendicular to a given line, or finds a perpendicular vector. In analytic geometry, perpendicularity is a basic relation: two nonzero lines are perpendicular when their slopes are negative reciprocals. This calculator automates that algebraic step, handles vertical/horizontal special cases, and produces point-slope, slope-intercept, or standard form equations suitable for plotting, proofs, design checks, and educational exercises.
About the Parallel and Perpendicular Line Calculator
The calculator accepts common line representations — slope-intercept \( y = mx + b \), point-slope \( y - y_0 = m(x - x_0) \), standard form \( Ax + By + C = 0 \), or two-point input. It computes the perpendicular slope \( m_{\perp} \), builds the perpendicular line through a specified point, and can output equivalent forms. For vector applications, it computes a perpendicular vector for a 2D vector \( (a,b) \). The tool explicitly handles edge cases: when the original line is vertical (\( x = c \)) the perpendicular is horizontal (\( y = k \)), and vice versa. Formulas are presented in render-ready math markup so they'll display responsively across screen sizes.
How to Use this Parallel and Perpendicular Line Calculator
1. Enter the line (choose slope-intercept, two points, or standard form) or enter a vector.
2. Provide a point through which the perpendicular line should pass (if required).
3. Click Calculate. The calculator returns the perpendicular slope, the perpendicular line in point-slope and slope-intercept forms,
and an equivalent standard-form equation. For vector input it returns a perpendicular vector and verifies perpendicularity using the dot product.
4. Use the produced equation to plot the perpendicular line, check orthogonality in designs, or solve geometry problems.
Formulas (render-ready)
If original slope is \( m \), perpendicular slope is:
\( \; m_{\perp} = -\dfrac{1}{m} \; \) (for \( m \neq 0 \)).
Perpendicular line through point \( (x_0,y_0) \):
\( \; y - y_0 = m_{\perp}(x - x_0) \; \)
In vector terms, vector \( \mathbf{u} = (a,b) \) has a perpendicular vector:
\( \; \mathbf{u}_{\perp} = (-b, a) \; \)
Orthogonality test (dot product):
\( \; \mathbf{u}\cdot\mathbf{v} = 0 \Longrightarrow \text{u and v are perpendicular} \; \)
Examples of Parallel and Perpendicular Line Calculator
Example 1 — Given \( y = 2x + 3 \). Then \( m = 2 \), so \( m_{\perp} = -\tfrac{1}{2} \). Through point \( (1,4) \): \( y - 4 = -\tfrac{1}{2}(x - 1) \) → \( y = -\tfrac{1}{2}x + \tfrac{9}{2} \).
Example 2 — Given line \( x = 5 \) (vertical). Perpendicular is horizontal: \( y = k \) (constant), so a perpendicular through (5,2) is \( y = 2 \).
Example 3 — Vector \( (3,4) \). A perpendicular vector is \( (-4,3) \) and dot product \( 3(-4)+4(3)=0 \), confirming orthogonality.
Responsive formula notes
All formulas are provided in math markup that will scale and reflow when rendered by a math-rendering engine on the page. Keep equations in inline or display math mode to preserve readability on narrow screens and ensure responsive layout.
How do I find the slope of a perpendicular line?
Take the negative reciprocal of the original slope: if \( m \) is the original slope, the perpendicular slope is \( -1/m \) (when defined).
What if the original line is vertical or horizontal?
Vertical lines (\( x=c \)) have undefined slope; their perpendicular lines are horizontal (\( y=k \)). Horizontal lines (slope 0) have vertical perpendiculars.
How do I write the perpendicular line through a point?
Compute \( m_{\perp} \), then use point-slope form: \( y - y_0 = m_{\perp}(x - x_0) \) and convert to other forms if needed.
Can I use two points to find a perpendicular line?
Yes — first compute the slope from two points, take its negative reciprocal, then build the perpendicular through the chosen point.
How to verify two vectors are perpendicular?
Compute the dot product; if it equals zero, the vectors are perpendicular.
What is a perpendicular vector to (a,b)?
A perpendicular vector in 2D is \( (-b,a) \) (or \( (b,-a) \)); either is orthogonal to \( (a,b) \).
How do I convert the perpendicular equation to standard form?
Rearrange the point-slope or slope-intercept form to \( Ax + By + C = 0 \) by moving all terms to one side and clearing denominators.