Problem 52 - College Algebra

Question

Determinants are used to show that three points lie on the same line (are collinear). If $$\left|\begin{array}{lll}x_{1} & y_{1} & 1 \\\x_{2} & y_{2} & 1 \\\x_{3} & y_{3} & 1\end{array}\right|=0$$ then the points \(\left(x_{1}, y_{1}\right),\left(x_{2}, y_{2}\right),\) and \(\left(x_{3}, y_{3}\right)\) are collinear. If the determinant does not equal 0 , then the points are not collinear. Are the points \((-4,-6),(1,0),\) and \((11,12)\) collinear?

Step-by-Step Solution

Answer

The points (-4,-6), (1,0), and (11,12) are not collinear as the determinant of the matrix formed by their coordinates is not equal to zero.

Step 1: Setup the Matrix

Set up the matrix replacing \(x_1\), \(y_1\), \(x_2\), \(y_2\), \(x_3\), \(y_3\) with the coordinates of the points (-4,-6),(1,0), and (11,12) respectively. Hence, the matrix is \[ \left| \begin{array}{ccc} -4 & -6 & 1\ 1 & 0 & 1 \ 11 & 12 & 1 \end{array} \right| \]

Step 2: Calculate the Determinant

Next is to calculate the determinant of this matrix. Using the formula for a 3x3 matrix determinant, we get \[ -4*(0*1 - 1*12) - (-6)*(1*1 - 1*11) + 1*(1*12 - 0*11) = -48 + 60 + 12 = 24 \]

Step 3: Draw the Conclusion

Finally, the determinant is not equal to 0 (it's 24), so the points (-4,-6), (1,0), and (11,12) are not collinear

Key Concepts

Collinearity

Collinearity is a concept in geometry where three or more points are said to be collinear if they lie on the same straight line. Understanding whether points are collinear helps in applications like defining linear paths or determining geometric alignments.
To check for collinearity using determinants, you can place the coordinates of the points into a mathematical structure called a matrix. When the determinant of this matrix equals zero, it indicates the points are collinear.
In practical terms, if you imagine drawing a straight line through each point on a graph, collinear points will all fall on that line without deviation. This can simplify understanding the arrangement and relation of points in space.

Linear Algebra

Linear Algebra is the branch of mathematics focused on vectors, vector spaces (also called linear spaces), linear transformations, and systems of linear equations. It provides the theoretical framework that supports understanding many geometric concepts, including collinearity.
A key concept in linear algebra is the matrix, which is a rectangular array of numbers. Matrices can represent systems of equations, transformations, and more. The determinant of a matrix, in particular, is a value that can offer insights into the properties of the matrix and the system it represents.
The collinearity of points using determinants is conceptually a part of linear algebra, linking geometric ideas with matrix computations.

Matrix Determinant

A matrix determinant is a special number calculated from a square matrix, like a 2x2 or 3x3 matrix. It provides crucial information about the matrix, such as whether it is invertible and if certain geometric properties hold.
For a 3x3 matrix, the determinant can be found using a specific formula:

• Multiply numbers diagonally and subtract cross products.
• It involves calculations like \(-4 imes (0 imes 1 - 1 imes 12)\) for one term of the determinant.

The determinant tells us something about the spatial relationship of points, such as whether they are collinear. If this determinant equals zero, it signals those points all lie on a single line in space.

3x3 Matrix

A 3x3 matrix is an arrangement of numbers across three rows and three columns. This specific type of matrix is frequently used in various computations involving three-dimensions, such as checking the collinearity of points.

• The matrix uses coordinates \(x_1, y_1\) through \(x_3, y_3\) to fill in its values.
• An extra column of ones is used to complete the third column, essential for correct calculation of properties like collinearity.

Interacting with these matrices involves operations like calculating determinants or performing row manipulations, foundational tasks in linear algebra and geometry.

Coordinate Geometry

Coordinate Geometry, also known as analytic geometry, involves using coordinates and algebra to solve geometric problems and describe geometric shapes.
This method lets you analyze and solve problems using a coordinate system, allowing for a clear and visual way to demonstrate concepts like distance, slope, and collinearity.
By placing geometric figures on a coordinate grid, complex calculations are simplified into algebraic terms. This approach also allows you to relate geometrical shapes and linear paths to algebraic equations, facilitating deeper mathematical insights and problem-solving strategies.