Perpendicular Line Calculator: The Slope and the Equation

How to use the perpendicular line calculator

• From a Slope. The slope box takes more than a number. Type a slope such as 2 or -2/3, or paste the line itself: y = 2x + 3, 3x + 4y = 12, y = 5, x = 4. For a vertical line you can also just type undefined.
• Two Points. Give two points the original line passes through and the slope is worked out first.
• Through x and Through y are the point the new line has to pass through. Leave them empty and the line is placed through the origin.

What you get back

• The perpendicular line, written in point slope form and solved for y. Most people searching for the perpendicular slope actually need the line, so it is printed without asking.
• The parallel line through the same point, since the two are almost always set in the same question.
• Both slopes, with fractions kept as fractions. The negative reciprocal of a whole number is always a fraction, and a fraction is the form the answer key expects.
• A real answer for horizontal and vertical lines, rather than an error. See below, because this is the case most tools get wrong.
• The check, m × m⊥ = −1, written into the steps so you can see the answer verify itself.

The rule, and the one case where it breaks

For two lines that both have a slope, the rule is short. Parallel lines have the same slope. Perpendicular lines have slopes that multiply to −1, which is the same as saying each is the negative reciprocal of the other: flip the fraction and change the sign.

So a line with slope 2 has perpendicular slope −1/2. A line with slope −2/3 has perpendicular slope 3/2. Flip, then change sign, or change sign, then flip. The order does not matter and you get the same answer either way.

Then there is the pair the rule cannot touch. A horizontal line has slope 0. A vertical line has no slope at all, because working it out means dividing by a run of zero. They are perpendicular to each other, and you can see that on any graph, but the negative reciprocal rule cannot produce that answer: −1 divided by 0 is undefined, and there is no number to take the reciprocal of on the vertical side.

The rule and the exception together:

• Perpendicular to y = 5 through (2, 7) is x = 2. Slope undefined.
• Perpendicular to x = 4 through (1, 3) is y = 3. Slope exactly 0.
• Everything else uses the negative reciprocal.

This calculator answers both of those. It is worth checking whichever tool you normally use against them, because in our review of the calculators currently ranking for this search, carried out in August 2026, several would not take x = 4 as an input at all, and one returned its answer in a box that can only hold y = mx + b, a form that cannot express a vertical line even in principle.

Two things to keep straight. Undefined is not infinity. A vertical line does not have a slope of −∞ or +∞; it has no slope, because the calculation never produces a number. And 0 and undefined are not interchangeable: the most common wrong answer to “what is the slope perpendicular to a horizontal line” is 0, which is the slope of the horizontal line you started with.

Why the slopes multiply to −1

Take a line with slope m, and read it as a right triangle: move 1 across, rise m up. Now rotate that triangle a quarter turn. The side that pointed right now points up, and the side that pointed up now points left. The run of 1 has become a rise of 1, and the rise of m has become a run of −m.

The rotated line therefore has slope 1 ÷ (−m), which is −1/m. Multiply the two together and the m cancels: m × (−1/m) = −1.

The rotation is also why the exception exists. Rotate a horizontal line and you get a vertical one, and at that quarter turn the run becomes 0, so the division that defines slope has nothing to divide by. The geometry still works. Only the arithmetic runs out. The same negative reciprocal appears again in calculus as the normal line, which is the line perpendicular to a curve at a point and is printed by the slope of tangent line calculator.

Perpendicular to a line in standard form

Textbook questions often give the line as Ax + By = C rather than solved for y, and it is quicker than rearranging. The slope of Ax + By = C is −A/B, so the perpendicular slope is B/A.

For 3x + 4y = 12: the slope is −3/4, so the perpendicular slope is 4/3. Through the origin that gives y = (4/3)x. You can paste the standard form straight into the slope box above and skip the rearranging.

The edge case is worth knowing. If B = 0 the line is Ax = C, which is vertical, so the perpendicular is horizontal and B/A does not apply.

Fractions, signs, and where the marks get lost

Two slips account for most wrong answers here, and neither is conceptual.

• Losing the fraction. The perpendicular slope of 2 is −1/2, and a calculator that hands back −0.5 has given you extra work at best and a mark deduction at worst. This one keeps the fraction.
• Sign tracking on a slope that is already negative. Starting from −2/3: flip it to −3/2, then change the sign to get +3/2. Two negatives were involved and the answer is positive. Where both slopes are numbers and neither is zero, they always have opposite signs, so if both of yours came out negative, something went wrong. The horizontal and vertical pair sits outside this: 0 and undefined have no signs to compare.

Checking your answer

Multiply the two slopes. If both lines have a slope and the product is −1, they are perpendicular. If the product is anything else, they are not.

All of this assumes two straight lines. Perpendicular to a curve means perpendicular to the curve’s tangent at the point you care about, so the first job there is finding that tangent’s slope, which the tangent line calculator does; the negative reciprocal step afterwards is the same one as above.

If one of the lines is vertical the test does not apply, and you check it by looking instead: a vertical line is perpendicular to a horizontal one and to nothing else. The steps panel prints the multiplication whenever both slopes exist, so the check is already done for you.