Build normal equations from points, slopes, and lines. Inspect intercepts, standard form, and plotted geometry. Export neat reports instantly for classwork, practice, and revision.
This page keeps a single-column layout. The calculator grid changes to three, two, or one columns by screen size.
| Case | Inputs | Computed Normal Slope | Normal Equation |
|---|---|---|---|
| Tangent slope mode | Point (2, 3), tangent slope 2 | -0.5 | y - 3 = -0.5(x - 2) |
| Slope-intercept mode | Original line y = 4x + 1, point (1, 5) | -0.25 | y - 5 = -0.25(x - 1) |
| Standard form mode | 2x - 3y + 6 = 0, point (4, 2) | -1.5 | y - 2 = -1.5(x - 4) |
| Horizontal source | Original line y = 7, point (3, 1) | Undefined | x = 3 |
If the source slope is m, then the normal slope is -1/m. This works whenever the source line is neither horizontal nor vertical.
A horizontal source line gives a vertical normal x = x0. A vertical source line gives a horizontal normal y = y0.
Once the normal slope is known, use y - y0 = mn(x - x0). The point (x0, y0) is the point supplied in the form.
For Ax + By + C = 0, the source slope is -A/B when B is not zero. The normal slope becomes B/A when A and B are non-zero.
A normal line is perpendicular to a given line or tangent. It passes through a chosen point, often the same point where the tangent touches a curve.
Take the negative reciprocal of the source slope. If the source slope is m, the normal slope is -1/m. Special horizontal and vertical cases are handled separately.
A horizontal source has slope zero. Its normal is vertical, so the equation becomes x = x0, where x0 is the point entered in the calculator.
A vertical source has undefined slope. Its normal is horizontal, so the equation becomes y = y0. The calculator detects this case automatically in standard form mode.
Different textbooks and teachers prefer different line forms. Showing point-slope, slope-intercept, and standard form helps you copy the version required for homework or revision.
Yes. The inputs accept decimals, negatives, and integers. You can also change decimal precision, which is useful when slopes or intercepts produce repeating decimals.
The graph plots the source line, the calculated normal line, and the reference point. This makes the right-angle relationship much easier to inspect visually.
CSV export saves your recent session history. PDF export saves the current result summary, including the input mode, source equation, normal forms, and key intercept details.
Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.