Determining Collinearity in Mathematics: Slopes and Determinants

Collinear

In mathematics, points that lie on the same straight line are said to be collinear

In mathematics, points that lie on the same straight line are said to be collinear. To determine if three points are collinear, we can use the concept of slope.

Let’s consider three points A, B, and C. These points are collinear if the slope of the line passing through AB is equal to the slope of the line passing through BC.

The slope of a line passing through two points (x1, y1) and (x2, y2) can be calculated using the formula:

slope = (y2 – y1) / (x2 – x1)

So, if the slope AB is equal to the slope BC, then the points A, B, and C are collinear.

Additionally, we can use the determinant method to check for collinearity. To do this, we arrange the coordinates of the three points into a 3×3 matrix, as follows:

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

If the determinant of this matrix is zero, then the points are collinear. If the determinant is not zero, the points are not collinear.

For example, consider the points A(2, 3), B(4, 6), and C(6, 9). Let’s check if these points are collinear.

Using the slope formula, we can calculate the slopes:

slope AB = (6-3) / (4-2) = 3/2
slope BC = (9-6) / (6-4) = 3/2

Since the slopes are equal, this indicates that the points A, B, and C are collinear.

Now let’s use the determinant method to verify this:

| 2 3 1 |
| 4 6 1 |
| 6 9 1 |

Determinant = (2*6*1) + (3*1*6) + (1*4*9) – (1*6*1) – (3*4*1) – (2*9*1) = 0

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

Remember that collinearity is not limited to just three points; any number of points can be collinear as long as the slopes or determinants are consistent.

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