Practice: Testing for Collinear Points Using Determinants

In a surveying project, a technician needs to verify that three markers at coordinates $(x_1, y_1)$, $(x_2, y_2)$, and $(x_3, y_3)$ are collinear (perfectly aligned on a straight line). They set up the following $3 \times 3determinant calculation:\begin{vmatrix} x_1 & y_1 & 1 \ x_2 & y_2 & 1 \ x_3 & y_3 & 1 \end{vmatrix}$

To confirm that the markers are collinear, what must the evaluated value of this determinant be?

A surveyor is verifying whether three marking stakes placed at different map coordinates (x, y) are perfectly aligned on a straight line (collinear). To do this, the surveyor uses a ${}3 imes 3$ determinant calculation. Match each component of the determinant setup and the final result to its correct function or meaning.

A structural engineer is verifying the alignment of three support pillars on a construction site. To confirm that the pillars are collinear (perfectly aligned in a straight line), the engineer evaluates a ${}3 imes 3$ determinant formed by the pillars' coordinates. If the pillars are indeed collinear, the evaluated value of this determinant must be ${}0$.

A land surveyor is using a coordinate system to verify if three markers placed at $(5, -5)$, $(4, -3)$, and $(3, -1)$ are perfectly aligned on a straight line. Arrange the following steps in the correct order to test for collinearity using the determinant method.

Matrix Setup for Collinearity Testing

A municipal surveyor is mapping a new utility line and needs to verify if three markers placed at coordinates $(x_1, y_1)$, $(x_2, y_2)$, and $(x_3, y_3)$ on a blueprint are perfectly aligned in a straight line (collinear). To test this using the determinant method, the surveyor sets up the following matrix equation:

$\begin{vmatrix} x_1 & y_1 & c x_2 & y_2 & c x_3 & y_3 & c \end{vmatrix} = 0$

To complete this ${}3 \times 3$ determinant setup correctly, the constant value of $c$ in the third column must be ____.

Verifying Alignment of Telecommunications Towers