Check if two vectors are collinear or not

Given six integers representing the x, y, and z coordinates of two vectors, the task is to check if the two given vectors are collinear or not.

Examples:

Input: x1 = 4, y1 = 8, z1 = 12, x2 = 8, y2 = 16, z2 = 24
Output: Yes
Explanation: The given vectors: 4i + 8j + 12k and 8i + 16j + 24k are collinear.

Input: x1 = 2, y1 = 8, z1 = -4, x2 = 4, y2 = 16, z2 = 8
Output: No
Explanation: The given vectors: 2i + 8j – 4k and 4i + 16j + 8k are not collinear.

Approach: The problem can be solved based on the idea that two vectors are collinear if any of the following conditions are satisfied:

• Two vectors A and B are collinear if there exists a number n, such that A = n · b.
• Two vectors are collinear if relations of their coordinates are equal, i.e. x1 / x2 = y1 / y2 = z1 / z2. 
  Note: This condition is not valid if one of the components of the vector is zero.
• Two vectors are collinear if their cross product is equal to the NULL Vector.

Therefore, to solve the problem, the idea is to check if the cross-product of the two given vectors is equal to the NULL Vector or not. If found to be true, then print Yes. Otherwise, print No. 

Below is the implementation of the above approach:

C++14

// C++ program for the above approach
#include <bits/stdc++.h>
using namespace std;

// Function to calculate cross
// product of two vectors
void crossProduct(int vect_A[],
                  int vect_B[],
                  int cross_P[])
{
    // Update cross_P[0]
    cross_P[0]
        = vect_A[1] * vect_B[2]
          - vect_A[2] * vect_B[1];

    // Update cross_P[1]
    cross_P[1]
        = vect_A[2] * vect_B[0]
          - vect_A[0] * vect_B[2];

    // Update cross_P[2]
    cross_P[2]
        = vect_A[0] * vect_B[1]
          - vect_A[1] * vect_B[0];
}

// Function to check if two given
// vectors are collinear or not
void checkCollinearity(int x1, int y1,
                       int z1, int x2,
                       int y2, int z2)
{
    // Store the first and second vectors
    int A[3] = { x1, y1, z1 };
    int B[3] = { x2, y2, z2 };

    // Store their cross product
    int cross_P[3];

    // Calculate their cross product
    crossProduct(A, B, cross_P);

    // Check if their cross product
    // is a NULL Vector or not
    if (cross_P[0] == 0 && cross_P[1] == 0
        && cross_P[2] == 0)
        cout << "Yes";
    else
        cout << "No";
}

// Driver Code
int main()
{
    // Given coordinates
    // of the two vectors
    int x1 = 4, y1 = 8, z1 = 12;
    int x2 = 8, y2 = 16, z2 = 24;

    checkCollinearity(x1, y1, z1,
                      x2, y2, z2);

    return 0;
}

Java

// Java program for the above approach
class GFG{

// Function to calculate cross
// product of two vectors
static void crossProduct(int vect_A[],
                         int vect_B[],
                         int cross_P[])
{

    // Update cross_P[0]
    cross_P[0] = vect_A[1] * vect_B[2] -
                 vect_A[2] * vect_B[1];

    // Update cross_P[1]
    cross_P[1] = vect_A[2] * vect_B[0] -
                 vect_A[0] * vect_B[2];

    // Update cross_P[2]
    cross_P[2] = vect_A[0] * vect_B[1] -
                 vect_A[1] * vect_B[0];
}

// Function to check if two given
// vectors are collinear or not
static void checkCollinearity(int x1, int y1,
                              int z1, int x2,
                              int y2, int z2)
{

    // Store the first and second vectors
    int A[] = { x1, y1, z1 };
    int B[] = { x2, y2, z2 };

    // Store their cross product
    int cross_P[] = new int[3];

    // Calculate their cross product
    crossProduct(A, B, cross_P);

    // Check if their cross product
    // is a NULL Vector or not
    if (cross_P[0] == 0 && cross_P[1] == 0 &&
        cross_P[2] == 0)
        System.out.print("Yes");
    else
        System.out.print("No");
}

// Driver Code
public static void main (String[] args)
{

    // Given coordinates
    // of the two vectors
    int x1 = 4, y1 = 8, z1 = 12;
    int x2 = 8, y2 = 16, z2 = 24;

    checkCollinearity(x1, y1, z1,
                      x2, y2, z2);
}
}

// This code is contributed by AnkThon

Python3

# Python3 program for the above approach

# Function to calculate cross
# product of two vectors
def crossProduct(vect_A, vect_B, cross_P):
    # Update cross_P[0]
    cross_P[0] = (vect_A[1] * vect_B[2] -
                  vect_A[2] * vect_B[1])

    # Update cross_P[1]
    cross_P[1] = (vect_A[2] * vect_B[0] -
                  vect_A[0] * vect_B[2])

    # Update cross_P[2]
    cross_P[2] = (vect_A[0] * vect_B[1] -
                  vect_A[1] * vect_B[0])

# Function to check if two given
# vectors are collinear or not
def checkCollinearity(x1, y1, z1, x2, y2, z2):

    # Store the first and second vectors
    A = [x1, y1, z1]
    B = [x2, y2, z2]

    # Store their cross product
    cross_P = [0 for i in range(3)]

    # Calculate their cross product
    crossProduct(A, B, cross_P)

    # Check if their cross product
    # is a NULL Vector or not
    if (cross_P[0] == 0 and
        cross_P[1] == 0 and
        cross_P[2] == 0):
        print("Yes")
    else:
        print("No")

# Driver Code
if __name__ == '__main__':

    # Given coordinates
    # of the two vectors
    x1 = 4
    y1 = 8
    z1 = 12
    x2 = 8
    y2 = 16
    z2 = 24

    checkCollinearity(x1, y1, z1, x2, y2, z2)

# This code is contributed by bgangwar59

C#

// C# program for the above approach
using System;

class GFG{

// Function to calculate cross
// product of two vectors
static void crossProduct(int []vect_A,
                         int []vect_B,
                         int []cross_P)
{

    // Update cross_P[0]
    cross_P[0] = vect_A[1] * vect_B[2] -
                 vect_A[2] * vect_B[1];

    // Update cross_P[1]
    cross_P[1] = vect_A[2] * vect_B[0] -
                 vect_A[0] * vect_B[2];

    // Update cross_P[2]
    cross_P[2] = vect_A[0] * vect_B[1] -
                 vect_A[1] * vect_B[0];
}

// Function to check if two given
// vectors are collinear or not
static void checkCollinearity(int x1, int y1,
                              int z1, int x2,
                              int y2, int z2)
{

    // Store the first and second vectors
    int []A = { x1, y1, z1 };
    int []B = { x2, y2, z2 };

    // Store their cross product
    int []cross_P = new int[3];

    // Calculate their cross product
    crossProduct(A, B, cross_P);

    // Check if their cross product
    // is a NULL Vector or not
    if (cross_P[0] == 0 && cross_P[1] == 0 &&
        cross_P[2] == 0)
        Console.Write("Yes");
    else
        Console.Write("No");
}

// Driver Code
public static void Main (string[] args)
{

    // Given coordinates
    // of the two vectors
    int x1 = 4, y1 = 8, z1 = 12;
    int x2 = 8, y2 = 16, z2 = 24;

    checkCollinearity(x1, y1, z1,
                      x2, y2, z2);
}
}

// This code is contributed by AnkThon

Javascript

<script>

        // Javascript program for the
        // above approach

        // Function to calculate cross
        // product of two vectors
        function crossProduct(vect_A,
            vect_B,
            cross_P) {
            // Update cross_P[0]
            cross_P[0]
                = vect_A[1] * vect_B[2]
                - vect_A[2] * vect_B[1];

            // Update cross_P[1]
            cross_P[1]
                = vect_A[2] * vect_B[0]
                - vect_A[0] * vect_B[2];

            // Update cross_P[2]
            cross_P[2]
                = vect_A[0] * vect_B[1]
                - vect_A[1] * vect_B[0];
        }

        // Function to check if two given
        // vectors are collinear or not
        function checkCollinearity(x1, y1,
            z1, x2,
            y2, z2) {
            // Store the first and second vectors
            let A = [x1, y1, z1];
            let B = [x2, y2, z2];

            // Store their cross product
            let cross_P = [];

            // Calculate their cross product
            crossProduct(A, B, cross_P);

            // Check if their cross product
            // is a NULL Vector or not
            if (cross_P[0] == 0 && cross_P[1] == 0
                && cross_P[2] == 0)
                document.write("Yes")
            else
                document.write("No")
        }

        // Driver Code

        // Given coordinates
        // of the two vectors
        let x1 = 4, y1 = 8, z1 = 12;
        let x2 = 8, y2 = 16, z2 = 24;

        checkCollinearity(x1, y1, z1,
            x2, y2, z2);


        // This code is contributed by Hritik

 </script>

Output

Yes

Time Complexity: O(1)
Auxiliary Space: O(1)

 Using the Cross Product:

Approach:

The cross product of two collinear vectors will be zero. We can use this property to check if two vectors are collinear or not.

Take input for the two vectors.
Calculate the cross product of the two vectors.
Check if the cross product is zero using the all() function.
If the cross product is zero, the vectors are collinear. Otherwise, they are not collinear

C++

#include <iostream>
using namespace std;

int main() {
    int x1 = 4, y1 = 8, z1 = 12;
    int x2 = 8, y2 = 16, z2 = 24;

    // Calculate the cross product using the formula (y1 * z2 - z1 * y2, z1 * x2 - x1 * z2, x1 * y2 - y1 * x2)
    int cross_product[3] = {y1 * z2 - z1 * y2, z1 * x2 - x1 * z2, x1 * y2 - y1 * x2};

    // Check if the cross product is zero
    if (cross_product[0] == 0 && cross_product[1] == 0 && cross_product[2] == 0) {
        cout << "Yes" << endl;
    } else {
        cout << "No" << endl;
    }

    return 0;
}

Java

public class CrossProductCheck {

    public static void main(String[] args) {
        int x1 = 4, y1 = 8, z1 = 12;
        int x2 = 8, y2 = 16, z2 = 24;

        // Calculate the cross product using the
        // formula (y1 * z2 - z1 * y2, z1 * x2 - x1 * z2, x1 * y2 - y1 * x2)
        int[] crossProduct = {
            y1 * z2 - z1 * y2,
            z1 * x2 - x1 * z2,
            x1 * y2 - y1 * x2
        };

        // Check if the cross product is zero
        if (crossProduct[0] == 0 && crossProduct[1] == 0 && crossProduct[2] == 0) {
            System.out.println("Yes");
        } else {
            System.out.println("No");
        }
    }
}

Python3

# input
x1, y1, z1 = 4, 8, 12
x2, y2, z2 = 8, 16, 24

# calculate cross product
cross_product = (y1 * z2 - z1 * y2, z1 * x2 - x1 * z2, x1 * y2 - y1 * x2)

# check if cross product is zero
if all(i == 0 for i in cross_product):
    print("Yes")
else:
    print("No")

C#

using System;

public class GFG
{
    public static void Main()
    {
        int x1 = 4, y1 = 8, z1 = 12;
        int x2 = 8, y2 = 16, z2 = 24;

        // Calculate the cross product using the formula (y1 * z2 - z1 * y2, z1 * x2 - x1 * z2, x1 * y2 - y1 * x2)
        int[] crossProduct = { y1 * z2 - z1 * y2, z1 * x2 - x1 * z2, x1 * y2 - y1 * x2 };

        // Check if the cross product is zero
        if (crossProduct[0] == 0 && crossProduct[1] == 0 && crossProduct[2] == 0)
        {
            Console.WriteLine("Yes");
        }
        else
        {
            Console.WriteLine("No");
        }
    }
}

Javascript

// Calculate the cross product using the formula (y1 * z2 - z1 * y2, z1 * x2 - x1 * z2, x1 * y2 - y1 * x2)
function calculateCrossProduct(x1, y1, z1, x2, y2, z2) {
    return [
        y1 * z2 - z1 * y2,
        z1 * x2 - x1 * z2,
        x1 * y2 - y1 * x2
    ];
}

// Check if the cross product is zero
function isZeroCrossProduct(crossProduct) {
    return crossProduct[0] === 0 && crossProduct[1] === 0 && crossProduct[2] === 0;
}

// Input values
let x1 = 4, y1 = 8, z1 = 12;
let x2 = 8, y2 = 16, z2 = 24;

// Calculate the cross product
let crossProduct = calculateCrossProduct(x1, y1, z1, x2, y2, z2);

// Check if the cross product is zero
if (isZeroCrossProduct(crossProduct)) {
    console.log("Yes");
} else {
    console.log("No");
}

Output

Yes

Time complexity: O(1)
Auxiliary Space: O(1)