Problem 60 - Precalculus

Question

three points, \(\left(x_{1}, y_{1}\right),\left(x_{2}, y_{2}\right),\) and \(\left(x_{3}, y_{3}\right),\) are collinear if and only if \(\left|\begin{array}{lll}x_{1} & y_{1} & 1 \\ x_{2} & y_{2} & 1 \\ x_{3} & y_{3} & 1\end{array}\right|=0\) Apply determinants to determine whether the points, \((2,-6),(-7,30),\) and \((5,-18),\) are collinear. Apply determinants to determine whether the points, \((2,-6),(-7,30),\) and \((5,-18),\) are collinear.

Step-by-Step Solution

Answer

The points \((2, -6), (-7, 30), (5, -18)\) are not collinear.

Step 1: Understand the problem

We need to determine if the given points \((2, -6), (-7, 30), (5, -18)\) are collinear. We will use the determinant method to check this.

Step 2: Form the determinant matrix

Create a 3x3 matrix by writing the coordinates of each point in the form: \[\begin{vmatrix}x_1 & y_1 & 1 \x_2 & y_2 & 1 \x_3 & y_3 & 1\end{vmatrix}\]For the given points, the matrix is:\[\begin{vmatrix}2 & -6 & 1 \-7 & 30 & 1 \5 & -18 & 1\end{vmatrix}\]

Step 3: Calculate the determinant

Use the formula for calculating the determinant of a 3x3 matrix: \[det = x_1(y_2z_3 - y_3z_2) - y_1(x_2z_3 - x_3z_2) + z_1(x_2y_3 - x_3y_2)\]In our case:\[det = 2(30 \times 1 - (-18) \times 1) - (-6)((-7) \times 1 - 5 \times 1) + 1((-7) \times (-18) - 5 \times 30)\]

Step 4: Simplify and solve the determinant

Calculating each term separately:- First term: \(2(30 + 18) = 2 \times 48 = 96\)- Second term: \(6(-7 - 5) = 6(-12) = -72\)- Third term: \(1(126 - 150) = 1(-24) = -24\)Now sum them: \[96 - (-72) - 24 = 96 + 72 - 24 = 144\]

Step 5: Interpret the result

If the determinant is zero, the points are collinear. Since we found that the determinant is \(144\), which is not zero, the points \((2, -6), (-7, 30), (5, -18)\) are not collinear.

Key Concepts

Collinear Points

Collinear points are three or more points that lie on a single straight line. To determine if points are collinear, you may utilize several methods, but a popular one in mathematics is the use of determinants.

When dealing with a 2D plane, you can check for collinearity by setting up a certain type of matrix—specifically a 3x3 matrix—based on the coordinates of the points.

• If the determinant of this matrix equals zero, the points are collinear.
• If it does not equal zero, the points are not collinear.

In practical terms, this means you could determine if points ae(2, -6), (-7, 30), and (5, -18) are collinear by computing their determinant and checking the result. The absence of a zero determinant verifies that these points do not lie on the same line.

3x3 Matrix

A 3x3 matrix is an arrangement of numbers in three rows and three columns. Such matrices are widely used in linear algebra to solve various equations and represent certain transformations in space.

The structure of a 3x3 matrix makes it useful for problems in physics, engineering, and graphics.

• Each cell (element) of the matrix represents a specific value or coefficient.
• In our case, the elements come from geometric point coordinates, with an extra column of 1's to form a closed matrix.

To find if points are collinear, arrange the coordinates into a 3x3 matrix: \[\begin{vmatrix}x_1 & y_1 & 1 \ x_2 & y_2 & 1 \ x_3 & y_3 & 1\end{vmatrix}\] This formation allows the use of determinants to verify linear alignment.

Matrix Determinant

The determinant of a matrix is a special number calculated from its elements, offering insight into the matrix properties, such as invertibility and, in our context, the collinearity of points.

For a 3x3 matrix, the determinant can be calculated using this method:

• Multiply each element of a row or column by the determinant of the sub-matrix obtained by deleting the corresponding row and column.
• Perform this for any of the rows or columns, often the first for simplicity, and combine with alternating signs.
• Resulting determinants sum up to form the total determinant value.

If this computed determinant is zero, the points are on a straight line. Otherwise, they are not collinear.

Linear Algebra

Linear algebra is a branch of mathematics dealing with vectors, vector spaces, and the operations applicable to them. The discipline is foundational for various fields such as computer science, physics, and engineering.

Among the many topics explored in linear algebra, matrices and determinants play a significant role.

• Matrices represent linear transformations and systems.
• Determinants provide criteria for matrix properties and, in applications such as ours, determine spatial relationships like collinearity of points.

Understanding determinants in linear algebra empowers you to solve problems about transformations, plane geometry, and more. This makes it an essential tool for examining the alignment and configuration of points in space.