#### 向量叉积的应用（三角形面积，线段相交，多边形面积，多边形凹凸性）

```//向量叉积的应用
#include <bits/stdc++.h>
#define EPS 1e-10
using namespace std;

struct point{
double x, y;
point(int x = 0, int y = 0) : x(x), y(y) {}
};

struct vect{
double x, y;
};

//获取由点a指向b的向量
vect get_vect(const point & a, const point & b)
{
vect res;
res.x = b.x - a.x;
res.y = b.y - a.y;
return res;
}

//计算两个向量的叉积
double chaji(const vect & v1, const vect & v2)
{
return v1.x * v2.y - v1.y * v2.x;
}

//计算三角形面积，a、b为三角形的两条边
double tri_area(const vect & a, const vect & b)
{
return abs(chaji(a, b))/2.0;
}

//判断点P是否在直线AB上
bool dot_in_line(const point & P, const point & A, const point & B)
{
vect v1 = get_vect(P, A);
vect v2 = get_vect(P, B);
return fabs(chaji(v1, v2))<EPS;
}

//判断两条线段P1P2,Q1Q2是否相交
bool is_intersect(const point & P1, const point & P2, const point & Q1, const point & Q2)
{
//P1P2跨立Q1Q2
vect v1 = get_vect(P1, Q1);
vect v2 = get_vect(Q2, Q1);
vect v3 = get_vect(P2, Q1);
double res1 = chaji(v1, v2) * chaji(v3, v2);
//Q1Q2跨立P1P2
vect u1 = get_vect(Q1, P1);
vect u2 = get_vect(P2, P1);
vect u3 = get_vect(Q2, P1);
double res2 = chaji(u1, u2) * chaji(u3, u2);
return res1<=0 && res2<=0;
}

//计算多边形的面积
double area(const vector<point> & dots)
{
vector<vect> vs;
for(size_t i = 1; i < dots.size(); i++)
vs.push_back(get_vect(dots.front(), dots[i]));
double sum = 0;
for(size_t i = 0; i < vs.size()-1; i++)
sum += tri_area(vs[i], vs[i+1]);
return sum;
}

//判断多边形的凹凸性
bool is_tu(const vector<point> & dots)
{
vector<vect> vs;
for(size_t i = 0; i < dots.size()-1; i++)
vs.push_back(get_vect(dots[i], dots[i+1]));
vs.push_back(get_vect(dots.back(), dots.front()));
vector<double> res;
for(size_t i = 0; i < vs.size()-1; i++)
res.push_back(chaji(vs[i], vs[i+1]));
res.push_back(chaji(vs.back(), vs.front()));
for(size_t i = 1; i < res.size(); i++)
if(!((res.front()>0 && res[i]>0)||(res.front()
}
```