Why is My Collinearity Check Function Returning Incorrect Results?

So, recently, I was working through a problem on CodeWars, and ran into a tricky situation. The task was to check if two vectors (x1, y1) and (x2, y2) are collinear (essentially, whether they lie on the same line passing through the origin in a 2D plane). Surprisingly, my code was returning the wrong results for the input (1, 2, 1, -2). The vectors (1, 2) and (1, -2) should not be collinear, but my function was returning true.

First, let’s take a look at the original code:

function collinearity(x1, y1, x2, y2) {
  switch (true) {
    case y1 % x1 === 0 && y2 % x2 === 0:
      return true;
    case x1 + x2 === 0 || (y1 === 0 && y2 === 0):
      return true;
    case x1 + y1 === 0 || x2 + y2 === 0:
      return true;
    default:
      return false;
  }
}

console.log(collinearity(1, 2, 1, -2)); // returns true, should be false

Understanding the Issue

The goal of the function is to determine whether the two vectors, represented by the coordinates, are collinear. The code checks several conditions, such as whether one vector is a scaled version of another or whether they negate each other to sum to zero. However, the logic was flawed.

Conditions Breakdown

1. Condition 1: y1 % x1 === 0 && y2 % x2 === 0

• This checks if both y-coordinates are multiples of their corresponding x-coordinates. However, this is not a proper check for collinearity.

1. Condition 2: x1 + x2 === 0 || (y1 === 0 && y2 === 0)

• This checks if the sum of the x-coordinates is zero (which would make the vectors horizontal opposites) or if both vectors lie along the x-axis. This doesn’t generally capture collinearity for all vectors.

1. Condition 3: x1 + y1 === 0 || x2 + y2 === 0

• This checks if the coordinate pairs sum to zero. Again, this doesn’t fully cover collinearity in a 2D plane.

Correct Approach to Check for Collinearity

To properly check for collinearity, we need to determine if the cross product of the two vectors is zero. The cross product of two vectors in a 2D plane ((v1 = (x1, y1)) and (v2 = (x2, y2))) is given by:

[ x1 * y2 – y1 * x2 ]

If this value is zero, the vectors are collinear.

Updated Solution

Let’s update the function with this approach:

function collinearity(x1, y1, x2, y2) {
  return (x1 * y2 - y1 * x2) === 0;
}

console.log(collinearity(1, 2, 1, -2)); // Should return false

In this solution, (x1 * y2 - y1 * x2) === 0 checks whether the cross product of the vectors is zero, which accurately determines if the vectors are collinear.

Explanation of the Corrected Code

1. Cross Product Calculation: The critical improvement is the use of the formula x1 * y2 - y1 * x2. This checks if the vectors lie on the same line.

1. Comparison: By comparing the cross product to zero, we can accurately determine collinearity without needing complex or misleading conditions.

Now, let’s run the corrected function:

console.log(collinearity(1, 2, 1, -2)); // Should return false

This time, the result is false, which correctly indicates that the vectors (1, 2) and (1, -2) are not collinear. This simple yet effective adjustment saves the day.

So, this was my journey in fixing a seemingly simple yet tricky issue. It turns out that understanding and applying the correct mathematical principles goes a long way in resolving such errors.