# Find area of triangle if two vectors of two adjacent sides are given

Given two vectors in form of (xi+yj+zk) of two adjacent sides of a triangle. The task is to find out the area of a triangle.
Examples:

Input:
x1 = -2, y1 = 0, z1 = -5
x2 = 1, y2 = -2, z2 = -1
Output: Area = 6.422616289332565
Input:
x1 = -2, y1 = 1, z1 = 5
x2 = 1, y2 = 3, z2 = -1
Output: Area = 8.860022573334675

Approach: Suppose we have two vectors a(x1*i+y1*j+z1*k) and b(x2*i+y2*j+z2*k) and we know that area of triangle is given by :

Area of triangle = (magnitude of cross product of vectors a and b) / 2 i.e |axb| / 2
And we know a X b = (y1*z2 – y2*z1)*i – (x1*z2 – x2*z1)*j + (x1*y2 – x2*y1)*k
Then area =

## C++

 // C++ program to calculate area of   // triangle if vectors of  // 2 adjacent sides are given   #include using namespace std ;   // function to calculate area of triangle   float area(int x1, int y1, int z1, int x2, int y2, int z2) {     float area = sqrt(pow((y1 * z2 - y2 * z1),2)               + pow((x1 * z2 - x2 * z1),2) +                pow((x1 * y2 - x2 * y1),2)) ;                area = area / 2;     return area ;       }       // Driver Code  int main() {     int    x1 = -2 ;     int y1 = 0 ;      int z1 = -5 ;     int x2 = 1 ;     int y2 = -2 ;     int z2 = -1 ;     float  a = area(x1, y1, z1, x2, y2, z2) ;       cout << "Area = " << a << endl;                               return 0; // This code is contributed by ANKITRAI1 }

## Java

 // Java program to calculate area of  // triangle if vectors of  // 2 adjacent sides are given import java.util.*;   class solution { // function to calculate area of triangle  static float area(int x1, int y1, int z1, int x2, int y2, int z2) {   double a =Math.pow((y1 * z2 - y2 * z1),2)              + Math.pow((x1 * z2 - x2 * z1),2) +                 Math.pow((x1 * y2 - x2 * y1),2); float area = (float)Math.sqrt(a) ;           area = area / 2;     return area ; }   //Driver program public static void main(String arr[]) {     int x1 = -2 ;     int y1 = 0 ;      int z1 = -5 ;     int x2 = 1 ;     int y2 = -2 ;     int z2 = -1 ;     float a = area(x1, y1, z1, x2, y2, z2) ;      System.out.println("Area= "+a); }   } //This code is contributed by //Surendra_Gangwar

## Python 3

 # Python code to calculate area of  # triangle if vectors of # 2 adjacent sides are given import math   # to calculate area of triangle  def area(x1, y1, z1, x2, y2, z2):     area = math.sqrt((y1 * z2 - y2 * z1) ** 2            + (x1 * z2 - x2 * z1) ** 2 +            (x1 * y2 - x2 * y1) ** 2)     area = area / 2     return area   # main function def main():     x1 = -2     y1 = 0     z1 = -5     x2 = 1     y2 = -2     z2 = -1     a = area(x1, y1, z1, x2, y2, z2)     print("Area = ", a)   # driver code     if __name__=="__main__":     main()

## C#

 // C# program to calculate area   // of triangle if vectors of  // 2 adjacent sides are given using System; class GFG { // function to calculate area of triangle  static float area(int x1, int y1, int z1,                    int x2, int y2, int z2) { double a = Math.Pow((y1 * z2 - y2 * z1), 2) +             Math.Pow((x1 * z2 - x2 * z1), 2) +            Math.Pow((x1 * y2 - x2 * y1), 2); float area = (float)Math.Sqrt(a) ;           area = area / 2;     return area ; }   // Driver Code public static void Main() {     int x1 = -2;     int y1 = 0;      int z1 = -5;     int x2 = 1;     int y2 = -2;     int z2 = -1;     float a = area(x1, y1, z1, x2, y2, z2);      Console.WriteLine("Area = " + a); } }   // This code is contributed // by inder_verma

## PHP

 

## Javascript

 

Output:

Area =  6.422616289332565

Time Complexity: O(logn) as it is using inbuilt sqrt function
Auxiliary Space: O(1), no extra space is required, so it is a constant.

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