Perpendicular Line Gradient Calculator

Perpendicular gradients, explained

In coordinate geometry, the gradient (slope) measures steepness. A line that rises 2 units for every 1 unit across has slope 2. A line that falls 3 units over 6 units has slope −3/6 = −1/2. Perpendicular lines meet at a right angle (90°), so their slopes are tightly linked.

The data rule: multiply to −1

For non-vertical lines, perpendicular gradients satisfy m1 × m2 = −1. So the perpendicular slope is the negative reciprocal: m_perp = −1/m. If m = 5/2 (2.5), then m_perp = −2/5 (−0.4). The calculator shows fraction and decimal side by side, clearly, with your chosen decimal places.

Horizontal and vertical exceptions

A horizontal line has slope 0, so a reciprocal is impossible. Its perpendicular is vertical, with an undefined slope. If the original line is vertical, the perpendicular is horizontal with slope 0. These cases are common in grid graphs, coordinate proofs, and rectangle or square problems.

Slope from two points

With points (x1, y1) and (x2, y2), use m = (y2 − y1)/(x2 − x1). Example: (1, 2) and (5, 10) give rise 8 and run 4, so m = 2. The perpendicular slope becomes −1/2. If x2 = x1, the original line is vertical, and the tool switches to the correct special case.

Slope from standard form

For Ax + By + C = 0, if B ≠ 0 then m = −A/B. For 3x + 6y − 12 = 0, m = −3/6 = −1/2, so m_perp = 2. This method is useful when a line is given from rearranged equations, constraints, or linear models in class.

Perpendicular line through a point

To build the full perpendicular line, add a point (x0, y0). Point-slope form is y − y0 = m_perp(x − x0). With m_perp = −1/2 through (4, 3), you get y − 3 = (−1/2)(x − 4). The calculator also outputs slope-intercept and standard form, so you can match whichever format your worksheet requests.

Quick checks and exports

Verify results by multiplying slopes to get −1, or by confirming vertical vs horizontal. A visual check helps too: a gentle upward slope has a steeper downward perpendicular. Export CSV for tables or PDF for submission. Keeping a record helps you compare multiple lines and spot sign mistakes.

FAQs

1) What does “perpendicular gradient” mean?

It is the slope of a line that meets the original line at 90 degrees. For non-vertical lines, it is the negative reciprocal, so multiplying the two slopes gives −1.

2) Why is the perpendicular slope −1/m?

Two perpendicular direction vectors have zero dot product. In slope form, that condition simplifies to m1 × m2 = −1, which rearranges to m2 = −1/m1 for defined slopes.

3) What happens if the original slope is 0?

If m = 0, the line is horizontal. Its perpendicular must be vertical, so the slope is undefined. The calculator reports this as an undefined (vertical) gradient.

4) How do I use two points to get the perpendicular slope?

First compute m = (y2 − y1)/(x2 − x1). Then take the negative reciprocal to get m_perp. If x2 equals x1, the original line is vertical and m_perp becomes 0.

5) Can I get the full perpendicular line equation?

Yes. Turn on the equation option and enter (x0, y0). The tool outputs point-slope, slope-intercept, and standard form so you can submit the format your teacher prefers.

6) Why does the calculator show fractions and decimals?

Fractions are exact and help with algebra. Decimals help with graphing and checking approximate steepness. Showing both reduces errors when signs, simplification, or rounding are involved.