Calculator Inputs
Choose a solving method, enter values, and generate the required vector equation or intersection line.
3D Plotly Graph
The graph shows the current plane or the current intersection line in three-dimensional space.
Formula Used
The vector equation of a plane uses one position vector and two independent spanning vectors.
r = r₀ + s u + t v
If you know a point and a normal vector, convert the normal information into a plane relation first.
n · (r - r₀) = 0
When three points define the plane, subtract one point from the other two to create spanning vectors.
u = B - A, v = C - A, n = u × v
For two planes, the line of intersection is parallel to the cross product of their normals.
d = n₁ × n₂, r = p + λd
A unit vector parallel to the intersection line is found by dividing the direction vector by its magnitude.
û = d / |d|
How to Use This Calculator
- Select the solving mode that matches the information you already have.
- Enter the required coordinates, plane coefficients, or vectors in the responsive input grid.
- Choose a precision value and graph scale if you want cleaner display output.
- Press Calculate Now to place the result above the form and below the header.
- Review the vector form, scalar form, parametric form, and derived normal or direction vectors.
- Use the export buttons to download a CSV summary or a PDF report.
Answers to Common Related Questions
Find a unit vector parallel to the line of intersection of the planes given by the equations
Take the normal vector from each plane, compute their cross product, and then normalize the result. If the normals are parallel, there is no unique intersection direction. Any normalized version of n₁ × n₂ gives a valid unit vector, up to sign.
Find an equation of the plane passing through the point perpendicular to the given vector or line
If a plane is perpendicular to a vector, that vector is the plane's normal. If a plane is perpendicular to a line, the line direction becomes the plane's normal. With point r₀ and normal n, use n · (r - r₀) = 0.
Equation of a plane through a point and perpendicular to a vector calculator
Use the Point and normal vector mode. Enter the point coordinates and the vector components. The calculator returns the vector equation, point-normal form, scalar equation, unit normal, and a 3D graph of the resulting plane.
Find the vector equation for the line of intersection of the planes
First find the line direction with n₁ × n₂. Then solve the two plane equations together to locate any point on both planes. Write the final line as r = p + λd, where p is that point and d is the direction.
Example Data Table
| Method | Input Example | Derived Output |
|---|---|---|
| Point and normal | P(1, 2, 3), n = (2, -1, 4) | Plane through P with normal n, plus generated spanning vectors |
| Three points | A(1, 0, 0), B(0, 1, 0), C(0, 0, 1) | u = B - A, v = C - A, then r = A + su + tv |
| Cartesian plane | 2x - y + 3z - 6 = 0 | Reference point on plane, normal vector, and vector equation |
| Plane intersection | x + y + z - 4 = 0 and 2x - y + z - 2 = 0 | Line direction from n₁ × n₂ and one common point |
Frequently Asked Questions
1. What is the vector equation of a plane?
It writes every point on a plane as one fixed point plus two independent direction vectors: r = r₀ + su + tv. The parameters s and t vary across all real numbers.
2. How do I find a unit vector parallel to the line of intersection of two planes?
Take the cross product of the two plane normals to get a direction vector. Divide that vector by its magnitude to obtain a unit vector. Either direction sign is acceptable.
3. How do I find an equation of a plane through a point and perpendicular to a vector?
Treat the given vector as the plane's normal. If the point is (x₀, y₀, z₀) and the vector is (a, b, c), use a(x-x₀)+b(y-y₀)+c(z-z₀)=0.
4. If a plane is perpendicular to a line, what vector becomes the normal?
The line's direction vector becomes the plane's normal vector. That lets you write the plane immediately if you also know one point on it.
5. How do I find the vector equation for the line of intersection of two planes?
Compute the direction using the cross product of the plane normals. Then solve the two plane equations simultaneously to get one common point. Combine them as r = p + λd.
6. Can three points always determine a plane?
Three points determine a unique plane only when they are not collinear. If they lie on one line, infinitely many planes can pass through them.
7. What happens if the two spanning vectors are parallel?
Parallel spanning vectors cannot define a plane because they create only one direction. The calculator checks this by testing whether their cross product is zero.
8. Can this calculator convert between vector, point-normal, and Cartesian forms?
Yes. Depending on the selected mode, it returns the vector equation and also shows equivalent point-normal, scalar, and parametric expressions whenever they are available.