# 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 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)

### 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 ` `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)

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