Equation of a Plane
Find the equation of a plane passing through three points in 3D space. Enter P1(x1,y1,z1), P2(x2,y2,z2), P3(x3,y3,z3) to compute the plane equation ax+by+cz+d=0 with the normal vector.
Enter three points in 3D space:
About This Calculator
The Equation of a Plane Calculator finds the unique plane passing through three points in 3D space. Using vector algebra, it computes two direction vectors from the points, finds the normal vector via cross product, and derives the standard plane equation ax + by + cz + d = 0.
This calculator is essential for multivariable calculus, linear algebra, 3D graphics programming, and engineering applications. The normal vector and plane equation are fundamental for computing distances, intersections, and reflections in 3D space.
Frequently Asked Questions
How do you find the equation of a plane through three points?
Given three points P1, P2, P3, compute two vectors u = P2-P1 and v = P3-P1. The normal vector n = u x v (cross product). The plane equation is a(x-x1) + b(y-y1) + c(z-z1) = 0, or ax + by + cz + d = 0 where d = -(a·x1 + b·y1 + c·z1).
What is the normal vector of a plane?
The normal vector is a vector perpendicular to the plane. It is computed as the cross product of two vectors lying in the plane. The components (a,b,c) of the normal vector become the coefficients in the plane equation ax + by + cz + d = 0.
Is this calculator free?
Yes, all calculators on Calculy are completely free to use.