Hallar el area de un triangulo si se dan dos vectores de dos lados adyacentes

Dados dos vectores en forma de (xi+yj+zk) de dos lados adyacentes de un triángulo. La tarea es encontrar el área de un triángulo.
Ejemplos: 
 

Entrada: 
x1 = -2, y1 = 0, z1 = -5 
x2 = 1, y2 = -2, z2 = -1 
Salida: Área = 6,422616289332565
Entrada: 
x1 = -2, y1 = 1, z1 = 5 
x2 = 1 , y2 = 3, z2 = -1 
Salida: Área = 8,860022573334675 
 

Enfoque: supongamos que tenemos dos vectores a(x1*i+y1*j+z1*k) y b(x2*i+y2*j+z2*k) y sabemos que el área del triángulo está dada por: 
 

Área del triángulo = (magnitud del producto vectorial de los vectores a y b) / 2, es decir, |axb| / 2 
Y sabemos a X b = (y1*z2 – y2*z1)*i – (x1*z2 – x2*z1)*j + (x1*y2 – x2*y1)*k Entonces área
= 

// C++ program to calculate area of 
// triangle if vectors of
// 2 adjacent sides are given
 
#include<bits/stdc++.h>
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 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 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# 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
// PHP program to calculate area
// of triangle if vectors of
// 2 adjacent sides are given
 
// function to calculate area of triangle
function area($x1, $y1, $z1,
              $x2, $y2, $z2)
{
    $area = sqrt(pow(($y1 * $z2 - $y2 * $z1), 2) +
                 pow(($x1 * $z2 - $x2 * $z1), 2) +
                 pow(($x1 * $y2 - $x2 * $y1), 2));
     
        $area = $area / 2;
    return $area ;
     
}
     
// Driver Code
$x1 = -2 ;
$y1 = 0 ;
$z1 = -5 ;
$x2 = 1 ;
$y2 = -2 ;
$z2 = -1 ;
$a = area($x1, $y1, $z1, $x2, $y2, $z2);
echo "Area = ".$a ."\n";                        
 
// This code is contributed by ChitraNayal
?>

<script>
 
// Javascript program to calculate area of 
// triangle if vectors of
// 2 adjacent sides are given
 
// function to calculate area of triangle 
function area( x1,  y1,  z1,  x2,  y2,  z2)
{
    let area = Math.sqrt(Math.pow((y1 * z2 - y2 * z1),2)
             + Math.pow((x1 * z2 - x2 * z1),2) +
               Math.pow((x1 * y2 - x2 * y1),2)) ;
     
         area = area / 2;
    return area ;
     
}
     
// Driver Code
 
    let    x1 = -2 ;
    let y1 = 0 ;
    let z1 = -5 ;
    let x2 = 1 ;
    let y2 = -2 ;
    let z2 = -1 ;
    let  a = area(x1, y1, z1, x2, y2, z2) ; 
          
    document.write( "Area= " +a);
 
// This code contributed by aashish1995
 
</script>

Area =  6.422616289332565

Complejidad de tiempo: O(1), el código se ejecutará en un tiempo constante.
Espacio auxiliar: O(1), no se requiere espacio adicional, por lo que es una constante.