#include<bits/stdc++.h>
#define ll long long
using namespace std;
int n;
const int N=1e5+5;
const ll inf=5e18;
struct Car{
	ll x;
	int y,id;
}a[N];
vector<pair<int,int>> G[N];
bool vis[N];
int sz[N],mx[N],all,rt;
int dep[N];
ll dis[N];
int dfn[N],mi[20][N],rnk[N],dfncnt;
int Fa[N];
int get(int x,int y){
	return dfn[x]<dfn[y]? x:y;
}
void init(int u,int fa){
	dep[u]=dep[fa]+1;
	mi[0][dfn[u]=++dfncnt]=fa;
	for(auto ed:G[u]){
		int v=ed.first,w=ed.second;
		if(v==fa)continue;
		dis[v]=dis[u]+w;
		init(v,u);
	}
	return;
}
void calc(int u,int fa){
	sz[u]=1,mx[u]=0;
	for(auto ed:G[u]){
		int v=ed.first;
		if(v==fa||vis[v])continue;
		calc(v,u);
		mx[u]=max(mx[u],sz[v]);
		sz[u]+=sz[v];
	}
	mx[u]=max(mx[u],all-sz[u]);
	if(!rt||mx[rt]>mx[u])rt=u;
	return;
}
void dfs(int u,int fa){
	Fa[u]=fa;
	vis[u]=1;
//	cout<<u<<' '<<fa<<endl;
	for(auto ed:G[u]){
		int v=ed.first;
		if(v==fa||vis[v])continue;
		rt=0;
		all=sz[v];
		calc(v,u);
		calc(rt,u);
		dfs(rt,u);
	}
	return;
}
ll dp[N];
struct Point{
	ll x,y;
	ll operator[](ll k){
		return y-x*k;
	}
};
struct LINE{
	vector<Point> stk;
	bool check(Point x,Point y,Point z){
		return 1.*(y.y-x.y)*(z.x-y.x)>1.*(z.y-y.y)*(y.x-x.x);
	}
	bool check(Point x,Point y,ll k){
		return 1.*(y.y-x.y)>1.*k*(y.x-x.x);
	}
	void add(Point x){
		//cout<<x.x<<' '<<x.y<<endl;
		int top=(int)stk.size();
		while(top>=1&&stk[top-1].y>x.y)--top;
		while(top>=2&&check(stk[top-2],stk[top-1],x)){
			--top;stk.pop_back();
		}
		stk.push_back(x);
		return;
	}
	ll query(ll k){
		if(stk.empty())return inf;
		int l=0,r=(int)stk.size()-2,res=(int)stk.size()-1;
		while(l<=r){
			int mid=l+r>>1;
			if(check(stk[mid],stk[mid+1],k))r=mid-1,res=mid;
			else l=mid+1;
		}
//		cout<<k<<' '<<res<<' '<<' '<<stk[res].x<<' '<<stk[res].y<<' '<<stk[res][k]<<' '<<check(stk[1],stk[2],k)<<'\n';
		return stk[res][k];
	}
}L[N];
int lca(int u,int v){
	if(u==v)return u;
	u=dfn[u],v=dfn[v];
	if(u>v)swap(u,v);
	int len=__lg(v-u++);
	return get(mi[len][u],mi[len][v-(1<<len)+1]);
}
ll get_dis(int x,int y){
	return dis[x]+dis[y]-dis[lca(x,y)]*2;
}
namespace PointmasTree{
	ll solve(int u){
		ll mn=inf;
		int x=u;
		while(x){
			mn=min(mn,L[x].query(get_dis(u,x)));
			x=Fa[x];
		}
		return mn;
	}
	void add(int u,ll X,ll Y){
		int x=u;
		while(x){
			L[x].add({-Y,dp[u]+X+Y*get_dis(u,x)});
			x=Fa[x];
		}
		return;
	}
}
vector<long long> travel(vector<long long> A, vector<int> B, vector<int> U,vector<int> V, vector<int> W){
	n=(int)A.size();
	for(int i=1;i<=n;i++){
		a[i].x=A[i-1],a[i].y=B[i-1],a[i].id=i;
	}
	sort(a+1,a+n+1,[&](Car a,Car b){
		return a.y>b.y;
	});
	for(int i=0;i<n-1;i++){
		G[U[i]+1].push_back({V[i]+1,W[i]});
		G[V[i]+1].push_back({U[i]+1,W[i]});
	}
	all=n,rt=0;
	init(1,0);
	calc(1,0);
	dfs(1,0);
	for(int j=1;(1<<j)<=n;j++){
		for(int i=1;i+(1<<j)-1<=n;i++){
			mi[j][i]=get(mi[j-1][i],mi[j-1][i+(1<<j-1)]);
		}
	}
	for(int i=2;i<=n;i++)dp[i]=inf;
	bool flag=0;
	for(int i=1;i<=n;i++){
		if(a[i].id!=1)dp[a[i].id]=PointmasTree::solve(a[i].id);
		else flag=1;
		if(flag)PointmasTree::add(a[i].id,a[i].x,a[i].y);
	}
//	cout<<'\n';
//	for(int i=1;i<=n;i++){
//		cout<<dp[i]<<' ';
//	}
//	cout<<'\n';
	vector<ll> ans;
	ans.resize(n-1,0);
	for(int i=2;i<=n;i++){
		ans[i-2]=PointmasTree::solve(i);
	}
	return ans;
}
#ifndef ONLINE_JUDGE
signed main(){
	int NN;
	scanf("%d",&NN);
	vector<ll> A;
	vector<int> B,U,V,W;
	for(int i=1;i<=NN;i++){
		ll x;
		scanf("%lld",&x);
		A.push_back(x);
	}
	for(int i=1;i<=NN;i++){
		int x;
		scanf("%d",&x);
		B.push_back(x);
	}
	for(int i=1,u,v,w;i<NN;i++){
		scanf("%d %d %d",&u,&v,&w);
		U.push_back(u);
		V.push_back(v);
		W.push_back(w);
	}
	vector<ll> ans=travel(A,B,U,V,W);
	for(int i=0;i<NN-1;i++)printf("%lld\n",ans[i]);
	return 0;
}
/*
input:
5
10 5 13 4 3
10 7 5 9 1
1 0 1
0 2 5
3 2 10
2 4 3

output:
20
60
104
88
*/
/*
input:
10
422984347234 607725142104 92118155366 486587956717 857158857503 799145939144 657596100723 796939845406 986487843217 23702497547
410761 491391 858697 690385 801672 657901 640039 950942 514882 665242
0 1 736069
1 2 871348
2 3 300958
3 4 249899
4 5 977539
5 6 867002
6 7 648549
7 8 891403
8 9 132040

output:
725332785743
1083248561571
1206870370609
1309519133748
1711054030927
2067184639449
2333583275238
2699736862921
2753973745361
*/
#endif