Math
Line of Intersection of Two Planes Calculator
Enter two planes in ax + by + cz = d or ax + by + cz + d = 0 form. mathcheck crosses the normal vectors to get the line's direction (shown component by component), solves for a point on the line by setting one coordinate to 0, and writes the parametric, symmetric, and vector equations.
Parallel planes and identical planes are reported instead of a meaningless line, and the angle between the planes is included.
Try an example
Result
Parametric equations
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- Symmetric equations
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- Direction vector n1 × n2
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- A point on the line
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- Unit direction
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- Angle between the planes
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- Relationship
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Student quick launch
Grade planning, algebra checks, and formulas students reach for most.
Study path
Use this calculator with
Follow these when you want the formula behind the answer, a short lesson, or nearby tools in the same topic.
Why two planes meet in a line
Two planes that are not parallel intersect along a straight line. That line lies in both planes, so it is perpendicular to both normal A quantity with both a size and a direction, often drawn as an arrow., and the cross product n1 × n2 points along it. Any single point that satisfies both plane equations then pins the line down.
2x + y - z = 4 and x - 3y + 2z = 1
n1 × n2 = ⟨(1)(2) - (-1)(-3), (-1)(1) - (2)(2), (2)(-3) - (1)(1)⟩ = ⟨-1, -5, -7⟩. Set z = 0: 2x + y = 4 and x - 3y = 1 give x = 13/7, y = 2/7. Line: x = 13/7 - t, y = 2/7 - 5t, z = -7t.
| n1 × n2 | Relationship | Intersection |
|---|---|---|
| non-zero | planes cross | a line |
| zero, different d | parallel planes | nothing |
| zero, same equation | identical planes | the whole plane |
How to use this calculator
- Enter the The number multiplying a variable, like the 3 in 3x. of both planes, choosing whether the constant sits on the right (= d) or on the left (+ d = 0).
- Read the parametric equations first, then the symmetric form, the direction vector, and the point the calculator chose.
- Use the angle between the planes if the problem asks how steeply they cross.
- Open Show the work to see each component of the cross product, the 2×2 system used for the point, and the check that the point lies on both planes.
How to read the answer
The direction vector is reduced by any common factor and may differ from yours by a scalar multiple; that is the same line. The point is found by setting the coordinate with the largest direction component to 0, so it is always a real point of the line. A 0 in the direction means the line is parallel to that coordinate plane, and the symmetric form then states that coordinate as a constant. Parallel or identical planes produce a message, not equations.
Common mistakes and edge cases
- Setting z = 0 to find a point when the direction has no z-component; then the line never reaches z = 0 and the 2×2 system has no solution. Choose a coordinate whose direction component is non-zero.
- Dropping the minus sign in the j component of the cross product.
- Declaring planes parallel because their d values differ; parallelism depends only on the normals.
- Writing the symmetric form with a 0 in a The bottom number of a fraction; it says how many equal parts make one whole.. Zero direction components must be written as an equation like y = 3 beside the rest.
- Assuming the calculator's point must match the book's; any point on the line works.
Worked examples
Textbook pair
2x + y - z = 4 and x - 3y + 2z = 1
Parametric equations
x = 13/7 - t, y = 2/7 - 5t, z = -7t
Direction with a zero component
x + y + z = 1 and x - y + z = 3
Parametric equations
x = t, y = -1, z = 2 - t
Coordinate planes
z = 0 and y = 0 meet along the x-axis
Parametric equations
x = -t, y = 0, z = 0
General-form input
x + 2y - z - 3 = 0 and 2x - y + 3z + 1 = 0
Parametric equations
x = t, y = 8/5 - t, z = 1/5 - t
Parallel planes
x + y + z = 1 and 2x + 2y + 2z = 5
Parametric equations
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Identical planes
x + y + z = 1 and 2x + 2y + 2z = 2
Parametric equations
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Decimal coefficients
0.5x + 0.5y = 1 and z = 0.5x
Parametric equations
x = -2t, y = 2 + 2t, z = -t
Not a plane
Parametric equations
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Frequently asked questions
How do I find the line of intersection of two planes?+
Cross the two normal vectors to get the direction, then find any point that satisfies both equations (set one coordinate to 0 and solve the 2×2 system). Combine them as x = x0 + d1·t, y = y0 + d2·t, z = z0 + d3·t.
Why is the direction the cross product of the normals?+
The line lies in both planes, so it is perpendicular to both normals. The cross product is the one direction perpendicular to two given vectors.
What if the cross product is the zero vector?+
The normals are parallel, so the planes are parallel. If the equations are multiples of each other the planes coincide; otherwise they never meet and there is no line.
My answer has a different point and direction. Is it wrong?+
Not necessarily. Any point on the line and any non-zero multiple of the direction describe the same line. Check by substituting your point into both plane equations and comparing direction ratios.
How do I write the symmetric equations when a direction component is 0?+
Leave that variable out of the chain and state it separately: for direction ⟨2, 0, 3⟩ through (1, 4, 0) write (x - 1)/2 = z/3, y = 4.
How is the angle between two planes defined?+
It is the angle between their normals, taken as the acute angle: cos θ = |n1 · n2| / (|n1| |n2|). Planes are perpendicular when n1 · n2 = 0.
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