Example: Testing for Collinear Points Using Determinants

A land surveyor is verifying that three boundary markers located at coordinates $(x_1, y_1)$, $(x_2, y_2)$, and $(x_3, y_3)$ are perfectly aligned in a straight line. According to the determinant test for collinearity, what must be the value of the following determinant?

$\begin{vmatrix} x_1 & y_1 & 1 \\ x_2 & y_2 & 1 \\ x_3 & y_3 & 1 \end{vmatrix}$

As an apprentice surveyor checking boundary markers, you use the determinant test to determine if three coordinate points form a perfectly straight fence line. According to this test, the points are considered collinear if the determinant of the matrix formed by their coordinates and a column of ones is equal to zero.

In the field of land surveying, technicians often use coordinate geometry to verify if three boundary markers are perfectly aligned in a straight line. This is done using a specific determinant test. Match the following components of the determinant test for collinearity with their corresponding mathematical roles or required values.

A quality control technician at a manufacturing plant is verifying that three drill holes on a metal plate at coordinates $(x_1, y_1)$, $(x_2, y_2)$, and $(x_3, y_3)$ are perfectly aligned in a straight line. According to the determinant test for collinearity, the determinant of the matrix shown below must be equal to ____ to confirm the holes are collinear.

$\begin{vmatrix} x_1 & y_1 & 1 \\ x_2 & y_2 & 1 \\ x_3 & y_3 & 1 \end{vmatrix}$

Alignment Verification in Precision Programming