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Example: Testing for Collinear Points Using Determinants
To determine whether three points are collinear, substitute their coordinates into a determinant with a column of ones. For example, to check the points , , and , evaluate: . Expanding by minors along the third column yields . Since the determinant equals , the points are collinear.
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Example: Testing for Collinear Points Using Determinants
A land surveyor is verifying that three boundary markers located at coordinates , , and are perfectly aligned in a straight line. According to the determinant test for collinearity, what must be the value of the following determinant?
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 , , and 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.
Alignment Verification in Precision Programming
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Practice: Testing for Collinear Points Using Determinants
In a surveying project, a technician needs to verify that three markers at coordinates , , and are collinear (perfectly aligned on a straight line). They set up the following $3 \times 3$ determinant calculation:
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 are perfectly aligned on a straight line (collinear). To do this, the surveyor uses a 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 determinant formed by the pillars' coordinates. If the pillars are indeed collinear, the evaluated value of this determinant must be .
A land surveyor is using a coordinate system to verify if three markers placed at , , and 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