# Program to find equation of a plane passing through 3 points

Given three points (x1, y1, z1), (x2, y2, z2), (x3, y3, z3). The task is to find the equation of the plane passing through these 3 points.

Examples:

Input: x1 = -1 y1 = w z1 = 1
x2 = 0 y2 = -3 z2 = 2
x3 = 1 y3 = 1 z3 = -4
Output: equation of plane is 26 x + 7 y + 9 z + 3 = 0.

Input: x1 = 2, y1 = 1, z1 = -1, 1
x2 = 0, y2 = -2, z2 = 0
x3 = 1, y3 = -1, z3 = 2
Output: equation of plane is -7 x + 5 y + 1 z + 10 = 0.

## Recommended: Please try your approach on {IDE} first, before moving on to the solution.

Approach: Let P, Q and R be the three points with coordinates (x1, y1, z1), (x2, y2, z2), (x3, y3, z3) respectively. Then the equation of plane is a * (x – x0) + b * (y – y0) + c * (z – z0) = 0, where a, b, c are direction ratios of normal to the plane and (x0, y0, z0) are co-ordinates of any point(i.e P, Q, or R) passing through the plane. For finding direction ratios of normal to the plane, take any two vectors in plane, let it be vector PQ, vector PR.

```=> Vector PQ = (x2 - x1, y2 - y1, z2 - z1) = (a1, b1, c1).
=> Vector PR = (x3 - x1, y3 - y1, z3 - z1) = (a2, b2, c2).
```

Normal vector to this plane will be vector PQ x vector PR.

```=> PQ X PR = (b1 * c2 - b2 * c1) i
+ (a2 * c1 - a1 * c2) j
+ (a1 * b2 - b1 *a2) k = ai + bj + ck.
```

Direction ratios of normal vector will be a, b, c. Taking any one point from P, Q, or R, let its co-ordinate be (x0, y0, z0). Then the equation of plane passing through a point(x0, y0, z0) and having direction ratios a, b, c will be

```=> a * (x - x0) + b * (y - y0) + c * (z - z0) = 0.
=> a * x - a * x0 + b * y - b * y0 + c * z - c * z0 = 0.
=> a * x + b * y + c * z + (- a * x0 - b * y0 - c * z0) = 0.
```

Below is the implementation of the above approach:

## C++

 `// C++ program to find equation of a plane  ` `// passing through given 3 points. ` `#include   ` `#include ` `#include ` `#include ` ` `  `using` `namespace` `std; ` ` `  `// Function to find equation of plane. ` `void` `equation_plane(``float` `x1, ``float` `y1,  ` `                    ``float` `z1, ``float` `x2, ` `                    ``float` `y2, ``float` `z2,  ` `                    ``float` `x3, ``float` `y3, ``float` `z3) ` `{ ` `    ``float` `a1 = x2 - x1; ` `    ``float` `b1 = y2 - y1; ` `    ``float` `c1 = z2 - z1; ` `    ``float` `a2 = x3 - x1; ` `    ``float` `b2 = y3 - y1; ` `    ``float` `c2 = z3 - z1; ` `    ``float` `a = b1 * c2 - b2 * c1; ` `    ``float` `b = a2 * c1 - a1 * c2; ` `    ``float` `c = a1 * b2 - b1 * a2; ` `    ``float` `d = (- a * x1 - b * y1 - c * z1); ` `    ``std::cout << std::fixed; ` `    ``std::cout << std::setprecision(2); ` `    ``cout << ``"equation of plane is "` `<< a << ``" x + "` `<< b ` `        ``<< ``" y + "` `<< c << ``" z + "` `<< d << ``" = 0."``; ` `} ` ` `  `// Driver Code  ` `int` `main() ` `{ ` `     `  `    ``float` `x1 =-1; ` `    ``float` `y1 = 2; ` `    ``float` `z1 = 1; ` `    ``float` `x2 = 0; ` `    ``float` `y2 =-3; ` `    ``float` `z2 = 2; ` `    ``float` `x3 = 1; ` `    ``float` `y3 = 1; ` `    ``float` `z3 =-4; ` `    ``equation_plane(x1, y1, z1, x2, y2, z2, x3, y3, z3); ` `    ``return` `0; ` `} ` `// This code is contributed  ` `// by Amber_Saxena. `

## C

 `// C program to find equation of a plane  ` `// passing through given 3 points. ` ` `  `#include ` ` `  `// Function to find equation of plane. ` `void` `equation_plane(``float` `x1, ``float` `y1,  ` `                    ``float` `z1, ``float` `x2, ` `                    ``float` `y2, ``float` `z2,  ` `                    ``float` `x3, ``float` `y3, ``float` `z3) ` `{ ` `    ``float` `a1 = x2 - x1; ` `    ``float` `b1 = y2 - y1; ` `    ``float` `c1 = z2 - z1; ` `    ``float` `a2 = x3 - x1; ` `    ``float` `b2 = y3 - y1; ` `    ``float` `c2 = z3 - z1; ` `    ``float` `a = b1 * c2 - b2 * c1; ` `    ``float` `b = a2 * c1 - a1 * c2; ` `    ``float` `c = a1 * b2 - b1 * a2; ` `    ``float` `d = (- a * x1 - b * y1 - c * z1); ` `    ``printf``(``"equation of plane is %.2f x + %.2f"` `        ``" y + %.2f z + %.2f = 0."``,a,b,c,d); ` `    ``return``; ` `} ` ` `  `// Driver Code ` `int` `main() ` `{ ` `    ``float` `x1 =-1; ` `    ``float` `y1 = 2; ` `    ``float` `z1 = 1; ` `    ``float` `x2 = 0; ` `    ``float` `y2 =-3; ` `    ``float` `z2 = 2; ` `    ``float` `x3 = 1; ` `    ``float` `y3 = 1; ` `    ``float` `z3 =-4; ` `    ``equation_plane(x1, y1, z1, x2, y2, z2, x3, y3, z3); ` `    ``return` `0; ` `} ` `// This code is contributed  ` `// by Amber_Saxena. `

## Java

 `// Java program to find equation  ` `// of a plane passing through  ` `// given 3 points. ` `import` `java .io.*; ` ` `  `class` `GFG ` `{ ` `     `  `// Function to find equation of plane. ` `static` `void` `equation_plane(``float` `x1, ``float` `y1,  ` `                           ``float` `z1, ``float` `x2, ` `                           ``float` `y2, ``float` `z2, ` `                           ``float` `x3, ``float` `y3, ` `                           ``float` `z3) ` `{ ` `    ``float` `a1 = x2 - x1; ` `    ``float` `b1 = y2 - y1; ` `    ``float` `c1 = z2 - z1; ` `    ``float` `a2 = x3 - x1; ` `    ``float` `b2 = y3 - y1; ` `    ``float` `c2 = z3 - z1; ` `    ``float` `a = b1 * c2 - b2 * c1; ` `    ``float` `b = a2 * c1 - a1 * c2; ` `    ``float` `c = a1 * b2 - b1 * a2; ` `    ``float` `d = (- a * x1 - b * y1 - c * z1); ` `    ``System.out.println(``"equation of plane is "` `+ a + ` `                       ``" x + "` `+ b + ``" y + "` `+ c +  ` `                       ``" z + "` `+ d + ``" = 0."``); ` `} ` ` `  `// Driver code ` `public` `static` `void` `main(String[] args) ` `{ ` `    ``float` `x1 =-``1``; ` `    ``float` `y1 = ``2``; ` `    ``float` `z1 = ``1``; ` `    ``float` `x2 = ``0``; ` `    ``float` `y2 =-``3``; ` `    ``float` `z2 = ``2``; ` `    ``float` `x3 = ``1``; ` `    ``float` `y3 = ``1``; ` `    ``float` `z3 =-``4``; ` `    ``equation_plane(x1, y1, z1, x2,  ` `                   ``y2, z2, x3, y3, z3); ` `} ` `} ` ` `  `// This code is contributed  ` `// by Amber_Saxena.  `

## Python

 `# Python program to find equation of a plane  ` `# passing through given 3 points. ` ` `  `# Function to find equation of plane. ` `def` `equation_plane(x1, y1, z1, x2, y2, z2, x3, y3, z3):  ` `     `  `    ``a1 ``=` `x2 ``-` `x1 ` `    ``b1 ``=` `y2 ``-` `y1 ` `    ``c1 ``=` `z2 ``-` `z1 ` `    ``a2 ``=` `x3 ``-` `x1 ` `    ``b2 ``=` `y3 ``-` `y1 ` `    ``c2 ``=` `z3 ``-` `z1 ` `    ``a ``=` `b1 ``*` `c2 ``-` `b2 ``*` `c1 ` `    ``b ``=` `a2 ``*` `c1 ``-` `a1 ``*` `c2 ` `    ``c ``=` `a1 ``*` `b2 ``-` `b1 ``*` `a2 ` `    ``d ``=` `(``-` `a ``*` `x1 ``-` `b ``*` `y1 ``-` `c ``*` `z1) ` `    ``print` `"equation of plane is "``, ` `    ``print` `a, ``"x +"``, ` `    ``print` `b, ``"y +"``, ` `    ``print` `c, ``"z +"``, ` `    ``print` `d, ``"= 0."` ` `  `# Driver Code  ` `x1 ``=``-``1`  `y1 ``=` `2` `z1 ``=` `1` `x2 ``=` `0` `y2 ``=``-``3` `z2 ``=` `2` `x3 ``=` `1` `y3 ``=` `1` `z3 ``=``-``4` `equation_plane(x1, y1, z1, x2, y2, z2, x3, y3, z3) `

## C#

 `// C# program to find equation  ` `// of a plane passing through  ` `// given 3 points. ` `using` `System;  ` ` `  `class` `GFG ` `{ ` `     `  `// Function to find equation of plane. ` `static` `void` `equation_plane(``float` `x1, ``float` `y1,  ` `                           ``float` `z1, ``float` `x2, ` `                           ``float` `y2, ``float` `z2, ` `                           ``float` `x3, ``float` `y3, ` `                           ``float` `z3) ` `{ ` `    ``float` `a1 = x2 - x1; ` `    ``float` `b1 = y2 - y1; ` `    ``float` `c1 = z2 - z1; ` `    ``float` `a2 = x3 - x1; ` `    ``float` `b2 = y3 - y1; ` `    ``float` `c2 = z3 - z1; ` `    ``float` `a = b1 * c2 - b2 * c1; ` `    ``float` `b = a2 * c1 - a1 * c2; ` `    ``float` `c = a1 * b2 - b1 * a2; ` `    ``float` `d = (- a * x1 - b * y1 - c * z1); ` `    ``Console.Write(``"equation of plane is "` `+ a + ` `                      ``"x + "` `+ b + ``"y + "` `+ c +  ` `                          ``"z + "` `+ d + ``" = 0"``); ` `} ` ` `  `// Driver code ` `public` `static` `void` `Main() ` `{ ` `    ``float` `x1 =-1; ` `    ``float` `y1 = 2; ` `    ``float` `z1 = 1; ` `    ``float` `x2 = 0; ` `    ``float` `y2 =-3; ` `    ``float` `z2 = 2; ` `    ``float` `x3 = 1; ` `    ``float` `y3 = 1; ` `    ``float` `z3 =-4; ` `    ``equation_plane(x1, y1, z1, ` `                   ``x2, y2, z2,  ` `                   ``x3, y3, z3); ` `} ` `} ` ` `  `// This code is contributed ` `// by ChitraNayal `

## PHP

 `  `

Output:

```equation of plane is  26 x + 7 y + 9 z + 3 = 0.
```

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Improved By : Amber_Saxena, chitranayal

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