Check if it is possible to reach vector B by rotating vector A and adding vector C to it

Given three 2-Dimentional vector co-ordinates A, B and C. The task is to perform below operations any number of times on vector A to get vector B :

  • Rotate the vector 90 degrees clockwise.
  • Add vector C to it.

Print “YES” B is obtained using the above operations, else Print “NO”.

Examples:

Input: Vector A: 2 3, Vector B: 2 3, Vector C: 0 0
Output: YES
The given vector A has coordinate (2, 3) and we need to 
convert this vector A to vector B which is also (2, 3). 
By rotating vector A 4 times by 90 degrees and adding
it to vector C(0, 0) will give us vector B(2, 3).

Input: Vector A: 0 0, Vector B: 1 1, Vector C: 2 2
Output: NO


Below is the step by step algorithm to solve this problem:

  • Initialize three vectors of 2-D coordinates as A ( a, b ), B ( x, y ) and C ( p, q ).
  • Coordinates of vector A can be of any quadrant. So, initialize a check function for all the quadrant and check if any of it is true.
  • Find a-x and b-y, which will tell us how much we need to make it to vector B.
  • Initialize d = p*p + q*q. If d = 0 then you need not to add anything in vector A.
  • If D > 0, then check if a*p + b*q and b*p – a*q is in the multiple of ‘d’ so that it is possible to get the vector B.

Below is the implementation of above algorithm:

C++

// C++ program to Check if it is
// possible to reach vector B by
// Rotating vector A and adding
// vector C to it any number of times
  
#include <bits/stdc++.h>
using namespace std;
#define ll long long
  
// function to check if vector B is
// possible from vector A
ll check(ll a, ll b, ll p, ll q)
{
    ll d = p * p + q * q;
  
    // if d = 0, then you need to add nothing to vector A
    if (d == 0)
        return a == 0 && b == 0;
    else
        return (a * p + b * q) % d == 0 && (b * p - a * q) % d == 0;
}
  
bool check(int a, int b, int x, int y, int p, int q)
{
    // for all four quadrants
    if (check(a - x, b - y, p, q)
        || check(a + x, b + y, p, q)
        || check(a - y, b + x, p, q)
        || check(a + y, b - x, p, q))
        return true;
    else
        return false;
}
  
// Driver code
int main()
{
    // initialize all three
    // vector coordinates
  
    int a = -4, b = -2;
    int x = 0, y = 0;
    int p = -2, q = -1;
  
    if (check(a, b, x, y, p, q))
        cout << "Yes";
    else
        cout << "No";
  
    return 0;
}

Java

// Java program to Check if it is 
// possible to reach vector B by 
// Rotating vector A and adding 
// vector C to it any number of times.
    
public class GFG {
  
    // function to check if vector B is 
    // possible from vector A 
    static boolean check(long a, long b, long p, long q) 
    
        long d = p * p + q * q; 
        
        // if d = 0, then you need to add nothing to vector A 
        if (d == 0
            return a == 0 && b == 0
        else
            return (a * p + b * q) % d == 0 && (b * p - a * q) % d == 0
    
        
    static boolean check(int a, int b, int x, int y, int p, int q) 
    
        // for all four quadrants 
        if (check(a - x, b - y, p, q) 
            || check(a + x, b + y, p, q) 
            || check(a - y, b + x, p, q) 
            || check(a + y, b - x, p, q)) 
            return true
        else
            return false
    
        
  
    // Driver code
    public static void main(String args[])
    {
        // initialize all three 
        // vector coordinates 
        
        int a = -4, b = -2
        int x = 0, y = 0
        int p = -2, q = -1
        
        if (check(a, b, x, y, p, q)) 
            System.out.println("Yes"); 
        else
            System.out.println("No"); 
      
    }
    // This Code is contributed by ANKITRAI1
}

Output:

Yes


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