[计算几何]判断四点共面

https://www.51nod.com/onlineJudge/questionCode.html#!problemId=1265
算三次叉积，判断模是否为0

#include<bits/stdc++.h>
using namespace std;
const double eps = 1e-10;
struct vec{
    double x, y, z;
};
double dot(const vec &a, const vec &b){ return a.x * b.x + a.y * b.y + a.z * b.z; }
vec cross(const vec &a, const vec &b){ return vec{a.y * b.z - a.z * b.y, a.z * b.x - a.x * b.z, a.x * b.y - a.y * b.x}; }
vec sub(const vec &a, const vec &b){ return vec{b.x - a.x, b.y - a.y, b.z - a.z}; }
bool iszero(const vec &a){ return sqrt(a.x * a.x + a.y * a.y + a.z * a.z) < eps ? 1 : 0; }
int main()
{
    ios::sync_with_stdio(0); cin.tie(0);
    int t;
    cin >> t;
    vec a, b, c, d;
    while (t--){
        cin >> a.x >> a.y >> a.z >> b.x >> b.y >> b.z >> c.x >> c.y >> c.z >> d.x >> d.y >> d.z;
        if (iszero(cross(cross(sub(a, d), sub(a, c)), cross(sub(a, c), sub(a, b))))) cout << "Yes" << endl;
        else cout << "No" << endl;
    }
    return 0;
}

一次叉积一次点积，直接判断结果是否为0，我也是醉了，怎么比上面那个还慢

#include<bits/stdc++.h>
using namespace std;
const double eps = 1e-10;
struct vec{
    double x, y, z;
};
double dot(const vec &a, const vec &b){ return a.x * b.x + a.y * b.y + a.z * b.z; }
vec cross(const vec &a, const vec &b){ return vec{a.y * b.z - a.z * b.y, a.z * b.x - a.x * b.z, a.x * b.y - a.y * b.x}; }
vec sub(const vec &a, const vec &b){ return vec{b.x - a.x, b.y - a.y, b.z - a.z}; }
int main()
{
    int t;
    cin >> t;
    vec a, b, c, d;
    while (t--){
        cin >> a.x >> a.y >> a.z >> b.x >> b.y >> b.z >> c.x >> c.y >> c.z >> d.x >> d.y >> d.z;
        if (fabs(dot(sub(a, b), cross(sub(a, c), sub(a, d)))) < eps)  cout << "Yes" << endl;
        else cout << "No" << endl;
    }
    return 0;
}