Math
Equation of a Plane Calculator
Give mathcheck three points, a point plus a normal vector, or a point plus two directions in the plane, and it returns the plane as ax + by + cz = d with cleared, reduced integer coefficients.
You also get the general form, the normal and unit normal vectors, the x-, y-, and z-intercepts, the point-normal, vector, and parametric forms, the distance from the origin, every cross-product step, and a sketch of the plane with its normal. Collinear points and parallel directions are reported instead of producing a bogus plane.
Try an example
Result
Plane (standard form)
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- General form
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- Normal vector
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- Unit normal
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- x-intercept
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- y-intercept
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- z-intercept
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More details (2 more)
- Parametric form
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- Distance from the origin
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Student quick launch
Grade planning, algebra checks, and formulas students reach for most.
Study path
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Follow these when you want the formula behind the answer, a short lesson, or nearby tools in the same topic.
How a plane is described
A plane in space is fixed by one point on it and a normal A quantity with both a size and a direction, often drawn as an arrow. n = ⟨a, b, c⟩ perpendicular to it. Any other point (x, y, z) is on the plane exactly when the vector from the known point to it is perpendicular to n, and writing that Multiply two vectors piece by piece and add the results to get one number. out gives the linear equation ax + by + cz = d.
Three ways to specify the plane
| Given | How the normal is found | Fails when |
|---|---|---|
| Three points P, Q, R | cross product of PQ and PR | the points are collinear or repeated |
| Point and normal | the normal is given | the normal is ⟨0, 0, 0⟩ |
| Point and two directions u, v | cross product u × v | u and v are parallel or zero |
Plane through (1, 2, 3), (3, -1, 2), (0, 1, 5)
PQ = ⟨2, -3, -1⟩ and PR = ⟨-1, -1, 2⟩. n = PQ × PR = ⟨(-3)(2) - (-1)(-1), (-1)(-1) - (2)(2), (2)(-1) - (-3)(-1)⟩ = ⟨-7, -3, -5⟩. Then d = n · P = -7 - 6 - 15 = -28, so -7x - 3y - 5z = -28, or 7x + 3y + 5z = 28 after making the leading coefficient positive.
How to use this calculator
- Choose what you know: three points, a point and a normal, or a point and two direction vectors.
- Enter the coordinates and components.
- Read the standard form in the header and the general form beneath it, then the normal, unit normal, A point where a graph crosses an axis., and distance from the origin.
- Copy the point-normal, vector, or parametric form from the formula list if your problem asks for one of those.
- Open Show the work to follow the cross product component by component and the substitution for d.
How to read the answer
The coefficients of x, y, and z are the components of a normal vector, so two planes are parallel exactly when their normals are multiples of each other. An intercept reads 'none' when the plane is parallel to that axis and 'contains the axis' when the axis lies inside the plane. d = 0 means the plane passes through the origin. The parametric form uses two parameters because a plane is two-dimensional.
Common mistakes and edge cases
- Crossing the wrong vectors. Both direction vectors must start at the same point: use Q - P and R - P, not Q - P and R - Q (that still works, but mixing a point with a direction does not).
- Forgetting the sign flip on the j component of the cross product.
- Solving for d with a point that is not on the plane.
- Reporting collinear points as a plane. If the cross product is ⟨0, 0, 0⟩, the points do not determine a plane.
- Reading the intercept of a plane parallel to an axis. If a = 0 and d ≠ 0 there is no x-intercept.
Worked examples
Three points
(1, 2, 3), (3, -1, 2), (0, 1, 5)
Plane (standard form)
7x + 3y + 5z = 28
Intercept form points
(1, 0, 0), (0, 1, 0), (0, 0, 1)
Plane (standard form)
x + y + z = 1
Decimal coordinates clear to integers
(0.5, 0, 0), (0, 0.5, 0), (0, 0, 1)
Plane (standard form)
2x + 2y + z = 1
Collinear points
(0, 0, 0), (1, 1, 1), (2, 2, 2)
Plane (standard form)
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Point and normal, common factor removed
P(1, 2, 3), n = ⟨2, -4, 6⟩
Plane (standard form)
x - 2y + 3z = 6
Plane through the origin parallel to the xy-plane's neighbor
P(0, 0, 0), n = ⟨0, 0, 5⟩ gives z = 0
Plane (standard form)
z = 0
Point and two directions
P(1, 2, 3), u = ⟨1, 0, -1⟩, v = ⟨0, 2, 1⟩
Plane (standard form)
2x - y + 2z = 6
Parallel directions
Plane (standard form)
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Zero normal
Plane (standard form)
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Frequently asked questions
How do I find the equation of a plane through three points?+
Form two vectors from one point to the other two, take their cross product to get a normal ⟨a, b, c⟩, then substitute one point into ax + by + cz = d to find d. The calculator shows each component of the cross product.
What is the normal vector of a plane?+
A vector perpendicular to the plane. For ax + by + cz = d the normal is ⟨a, b, c⟩, read straight from the coefficients. Any non-zero multiple is also a normal.
Why does the calculator change my coefficients?+
Multiplying an equation by a constant does not change the plane. The calculator divides out common factors, clears fractions from decimal inputs, and makes the leading coefficient positive so that 2x - 4y + 6z = 12 is reported as x - 2y + 3z = 6.
What does it mean if the three points are collinear?+
They lie on a single line, so there is no unique plane; infinitely many planes contain that line. The cross product comes out as the zero vector and the calculator reports the problem instead of an equation.
How do I write a plane in parametric form?+
Pick a point on the plane and two non-parallel direction vectors in it: (x, y, z) = P + s·u + t·v. From three points, u = Q - P and v = R - P work directly; from a normal, any two vectors perpendicular to n will do.
How do I find where a plane crosses the axes?+
Set the other two variables to 0. The x-intercept is d/a, the y-intercept d/b, and the z-intercept d/c. If a coefficient is 0 the plane is parallel to that axis and has no intercept there (unless d is also 0, in which case the axis lies in the plane).
About this calculator
- Written by
- mathcheck editorial team
- Last reviewed
- September 4, 2026
Method
- Uses the values entered by the user and stable formulas documented on the page.
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Last updated: September 4, 2026