What This Calculator Does
This tool finds the equation of a plane in three-dimensional space that passes through three given points A, B and C. The result is expressed in the standard general form ax + by + cz + d = 0, where (a, b, c) is a vector perpendicular (normal) to the plane and d locates the plane in space. Three non-collinear points uniquely determine a single plane, which is exactly what this calculator returns.
How To Use It
Enter the x, y and z coordinates of each of the three points. The values are plain real numbers (positive, negative or zero) — there are no units to convert. Press calculate and the tool returns the four coefficients a, b, c, d along with the fully formatted plane equation. If your three points happen to lie on a single line (or two of them coincide), no unique plane exists and the calculator will tell you so.
The Formula Explained
First build two edge vectors of the triangle: \vec{u} = B - A and \vec{v} = C - A. Their cross product \vec{n} = \vec{u} × \vec{v} gives a vector normal to the plane, and its components are the coefficients a, b and c:
Because point A lies on the plane, we solve for d by substitution: d = -(a· x_A + b· y_A + c· z_A). Any scalar multiple of the equation describes the same plane, so the raw cross-product coefficients are perfectly valid.
Worked Example
Take A = (1, 2, -2), B = (3, -2, 1), C = (5, 1, -4). Then \vec{u} = (2, -4, 3) and \vec{v} = (4, -1, -2). The cross product gives
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