Enter Line and Point Details
Use the standard form Ax + By + C = 0. The calculator creates the unique perpendicular line through your selected point.
Tip: A and B cannot both equal zero.
Example Data Table
| Input line | Point | Perpendicular line | Line type |
|---|---|---|---|
| 2x − 3y + 6 = 0 | (4, 1) | −3x − 2y + 14 = 0 | Sloped |
| x − 5 = 0 | (2, 7) | y − 7 = 0 | Vertical to horizontal |
| y + 4 = 0 | (3, 9) | x − 3 = 0 | Horizontal to vertical |
Formula Used
Input line: Ax + By + C = 0
Selected point: (x₀, y₀)
Perpendicular line: Bx − Ay + Ay₀ − Bx₀ = 0
Foot point: (x₀ − A(Ax₀ + By₀ + C)/(A² + B²), y₀ − B(Ax₀ + By₀ + C)/(A² + B²))
Shortest distance: |Ax₀ + By₀ + C| / √(A² + B²)
The perpendicular equation uses the original line’s normal vector as its direction. This guarantees a ninety-degree intersection.
How to Use This Calculator
- Enter A, B, and C. Copy the coefficients from your line in standard form.
- Enter the selected point. Add its x-coordinate and y-coordinate.
- Choose precision. Select the decimals shown in the result.
- Calculate the perpendicular line. The result appears above this form.
- Review the checks. Confirm the foot point, distance, and zero dot product.
- Export when needed. Download CSV data or use the print option to save a PDF.
Understanding Perpendicular Lines
A perpendicular line meets another line at a right angle. That angle measures ninety degrees. The relationship is simple. A calculator reduces manual sign mistakes. You start with a line written as Ax + By + C = 0. Choose a point. The point may lie on the original line. It may also sit away from it. In both cases, one unique perpendicular line passes through that point. These results support clear diagrams, repeatable work, and confident communication between teams using coordinate systems.
The input coefficients describe the original line. A and B define its orientation. C moves the line across the coordinate plane. The selected point supplies the new line location. The calculator combines these values without changing the geometry. It returns standard form first. That makes it more reliable than slope form alone.
Why the Coefficients Change
The original line has normal vector (A, B). A perpendicular line uses that normal vector as its direction vector. Its standard form can be written as Bx − Ay + Ay₀ − Bx₀ = 0. Here, x₀ and y₀ are the selected point coordinates. This expression always passes through the point. It also creates a right angle with the original line.
The calculator shows the working values. It evaluates Ax₀ + By₀ + C. This tells you whether the point already lies on the original line. A zero result means the point is on it. A nonzero result means it is away from it. The tool also finds the foot of the perpendicular. That is the exact intersection point. The shortest distance uses that same intersection.
Reading Slopes and Special Cases
For a nonvertical input line, its slope is −A divided by B. The perpendicular slope is B divided by A. Their product equals negative one when both slopes exist. This is a useful check. However, vertical lines do not have ordinary slopes. A vertical original line produces a horizontal perpendicular line. A horizontal original line produces a vertical perpendicular line.
Standard form avoids confusion in these special cases. It does not require dividing by zero. It still gives a complete equation. The direction vectors provide another check. Their dot product is zero. A zero dot product confirms perpendicular directions.
Useful Practical Checks
Enter coefficients with their correct signs. A missing minus sign changes the line. Use decimal values when measurements require them. Keep enough decimal places for scientific or design work. Review the result before copying it into another document. Check that the selected point satisfies the new equation. Substitute the point coordinates into the perpendicular result. The total should equal zero, aside from rounding.
The foot point should satisfy both equations. This provides a strong verification step. Distance is always nonnegative. A distance of zero means the selected point lies on the input line. Use the CSV file for spreadsheets or records. Use the print option when a saved calculation is needed. Repeat the calculation whenever the point or coefficients change. Accurate inputs create dependable perpendicular equations every single time.
Frequently Asked Questions
1. What does this calculator find?
It creates the equation of a line that passes through your selected point and forms a ninety-degree angle with the input line.
2. Which line format should I enter?
Enter the input line as Ax + By + C = 0. Put the x coefficient in A, the y coefficient in B, and the constant in C.
3. Can I enter decimal coefficients?
Yes. The calculator accepts decimal and negative values. Choose a suitable display precision when your input represents measurements or scientific data.
4. Why cannot A and B both be zero?
When A and B are both zero, the expression does not describe a line. At least one coefficient must be nonzero.
5. Must the selected point lie on the input line?
No. A unique perpendicular line can pass through any point. The calculator also reports whether your selected point lies on the original line.
6. Why does a vertical line show an undefined slope?
A vertical line has no horizontal change, so ordinary slope division is impossible. Its perpendicular line is horizontal and has slope zero.
7. What is the foot of the perpendicular?
It is the intersection point of the original line and the new perpendicular line. It is also the closest point on the original line.
8. Does rounding change the actual calculation?
The selected precision controls displayed decimals. The calculation uses floating-point values, so small display differences can occur after rounding.
9. Can different equations describe the same result?
Yes. Multiplying every coefficient by the same nonzero number creates an equivalent line. The geometry remains unchanged.
10. What does the CSV download include?
The file includes the input equation, point, perpendicular equation, slopes, foot point, distance, intercepts, and vector verification details.
11. How can I verify the result?
Substitute the point into the new equation. Confirm the total is zero. Accurate inputs create dependable perpendicular equations every single time.