Determinant Method: Testing Collinearity of Points in Mathematics

Noncollinear

In mathematics, noncollinear refers to a set of points that do not lie on the same straight line

In mathematics, noncollinear refers to a set of points that do not lie on the same straight line. Collinear points, on the other hand, are points that do lie on the same straight line.

To determine if a set of points is noncollinear, you can use a method called the “collinearity test.” There are a few different approaches to perform this test, but one commonly used method is known as the “determinant method.”

Let’s say we have three points A (x1, y1), B (x2, y2), and C (x3, y3). To check for collinearity, we can calculate the determinant of the matrix formed by the coordinates of these points:

| x1 y1 1 |
| x2 y2 1 |
| x3 y3 1 |

The determinant of this matrix is given by the formula:

Det = (x1 * y2 * 1 + y1 * 1 * x3 + 1 * x2 * y3) – (1 * y2 * x3 + x1 * 1 * y3 + y1 * x2 * 1)

If the determinant is zero, then the points A, B, and C are collinear. If the determinant is nonzero, then the points are noncollinear.

For example, let’s consider three points A(2, 3), B(4, 5), and C(6, 7). We can calculate the determinant as follows:

Det = (2 * 5 * 1 + 3 * 1 * 6 + 1 * 4 * 7) – (1 * 5 * 6 + 2 * 1 * 7 + 3 * 4 * 1)
= (10 + 18 + 28) – (30 + 14 + 12)
= 56 – 56
= 0

Since the determinant is zero, the points A, B, and C are collinear.

On the other hand, if the determinant resulted in a nonzero value, for instance, Det = 16, then the points A, B, and C would be noncollinear.

So, the collinearity test using the determinant method is an effective way to determine if a set of points is noncollinear.

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