Enter the slope of one line to find the slope of a line perpendicular to it.
Perpendicular Slope (m2): —
Understanding Perpendicular Slopes
In coordinate geometry, two lines are considered perpendicular if they intersect at a right angle (90 degrees). The relationship between their slopes is a fundamental concept.
The Mathematical Relationship
If you have two non-vertical lines with slopes m1 and m2, they are perpendicular if and only if the product of their slopes is -1. Mathematically, this is expressed as:
m1 * m2 = -1
This implies that the slope of one line is the negative reciprocal of the slope of the other line:
m2 = -1 / m1
Special Cases: Horizontal and Vertical Lines
A horizontal line has a slope of 0.
A vertical line has an undefined slope.
A line perpendicular to a horizontal line (slope 0) is a vertical line (undefined slope), and vice-versa. Our calculator handles these cases.
How to Input Slopes
You can enter the slope in a few common formats:
Decimal: Enter the slope as a decimal number (e.g., 2, -0.5).
Fraction: Enter the slope as a fraction (e.g., 1/2, -3/4). The calculator will parse this correctly.
Use Cases
The concept of perpendicular slopes is vital in various fields:
Construction and Architecture: Ensuring walls are perfectly vertical and foundations are level relies on perpendicularity.
Engineering: Designing structures, machinery, and circuits often involves perpendicular alignments for optimal function.
Computer Graphics: Calculating lighting, reflections, and object orientations uses geometric principles including perpendicularity.
Mathematics Education: Understanding this relationship is a cornerstone of analytic geometry.
Calculator Logic
This calculator takes the slope of one line (m1) as input. It then calculates the slope of the perpendicular line (m2) using the formula m2 = -1 / m1. It also handles the special cases where m1 is 0 (horizontal line, perpendicular is vertical/undefined) and if m1 is undefined (vertical line, perpendicular is horizontal/slope 0).
function parseSlope(slopeString) {
slopeString = slopeString.trim();
if (slopeString === "") return NaN; // Empty input
// Handle "undefined" or "infinite"
if (slopeString.toLowerCase() === 'undefined' || slopeString.toLowerCase() === 'infinite') {
return Infinity; // Represent undefined slope as Infinity
}
// Check for fraction format like "a/b"
var parts = slopeString.split('/');
if (parts.length === 2) {
var numerator = parseFloat(parts[0]);
var denominator = parseFloat(parts[1]);
if (!isNaN(numerator) && !isNaN(denominator)) {
if (denominator === 0) {
return Infinity; // Division by zero means undefined slope
}
return numerator / denominator;
}
}
// Try parsing as a regular number
var num = parseFloat(slopeString);
if (!isNaN(num)) {
return num;
}
return NaN; // Not a valid number or fraction
}
function formatSlope(slope) {
if (slope === Infinity) {
return "Undefined (Vertical Line)";
}
if (slope === -Infinity) {
return "Undefined (Vertical Line)";
}
if (slope === 0) {
return "0 (Horizontal Line)";
}
if (isNaN(slope)) {
return "Invalid Input";
}
// Try to represent as a simple fraction if possible
var tolerance = 1e-6;
var reciprocal = 1 / slope;
var num = Math.round(reciprocal);
var den = slope * num;
if (Math.abs(den – Math.round(den)) < tolerance) {
var roundedDen = Math.round(den);
if (roundedDen === 0) return slope.toFixed(4); // Avoid division by zero if slope is huge
return num + "/" + roundedDen;
}
// Fallback to decimal if fraction conversion is not clean
return slope.toFixed(4);
}
function calculatePerpendicularSlope() {
var slope1Input = document.getElementById("slope1").value;
var resultDiv = document.getElementById("result");
var m1 = parseSlope(slope1Input);
if (isNaN(m1)) {
resultDiv.textContent = "Perpendicular Slope (m2): Invalid Input";
resultDiv.style.color = "#dc3545"; // Red for error
return;
}
var m2;
if (m1 === 0) {
// m1 is 0 (horizontal line), perpendicular is vertical (undefined slope)
m2 = Infinity;
} else if (m1 === Infinity || m1 === -Infinity) {
// m1 is undefined (vertical line), perpendicular is horizontal (slope 0)
m2 = 0;
} else {
// General case: m2 = -1 / m1
m2 = -1 / m1;
}
resultDiv.textContent = "Perpendicular Slope (m2): " + formatSlope(m2);
resultDiv.style.color = "#28a745"; // Green for success
}