# Check if two vectors are collinear or not

• Difficulty Level : Basic
• Last Updated : 22 Apr, 2021

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 Â· 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 ``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

 ``
Output:
`Yes`

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

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