Problem 52 - Algebra and Trigonometry

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. Use this information to work Exercises \(51-52\) Are the points \((-4,-6),(1,0),\) and \((11,12)\) collinear?

Step-by-Step Solution

Answer

No, the points \((-4,-6),(1,0)\), and \((11,12)\) are not collinear as the determinant does not equal zero, it equals 46.

Step 1: Form the Matrix

Start by forming the 3x3 matrix as follows: \[ \left|\begin{array}{ccc}{-4} & {-6} & {1} \ {1} & {0} & {1} \ {11} & {12} & {1}\end{array}\right|.\] The x and y coordinates of the points are inserted into the first and second columns, respectively, and the third column is filled with 1s.

Step 2: Calculate the Determinant using Cofactor Expansion

The determinant of the matrix can be calculated using the expansion by minors (or cofactor expansion) method. This leads to the following: \[ -4 * det\left[ \begin{array}{cc} 0 & 1 \\ 12 & 1 \end{array} \right] + (-1) * det \left[\begin{array}{cc} 1 & 1 \\ 11 & 1 \end{array} \right] - 1 * det \left[\begin{array}{cc} 1 & 0 \\ 11 & 12 \end{array} \right]. \] The determinant of a 2x2 matrix \[ \begin{array}{cc} a & b \\ c & d \end{array} \] is calculated as ad - bc.

Step 3: Compute the Determinants and Add Them

Calculate the determinants from step 2 to get: \[ -4*(0-12) -1 * (1-11) - 1 * (12-0). \] After simplifying the expression, the result is -4*-12 -1*-10 -12.

Step 4: Evaluate the Determinant Value

Check if the determinant is equal to zero. If it is, then the points are collinear. Otherwise, the points are not collinear. The expression from step 3 simplifies to 48 +10 - 12. Calculating this gives a final answer of 46.

Key Concepts

Determinant Calculation

The determinant of a matrix is a special number that can be calculated from its elements. It helps to provide valuable information about the matrix, such as whether a set of points in the coordinate plane is collinear. To determine if three points are collinear, we form a 3x3 matrix using their coordinates, then calculate the determinant to see if the value is zero. If the determinant equals zero, the points are collinear. If not, they aren't on the same line.
The process involves plugging the x and y coordinates of the points into the rows and columns of the matrix, adding a column of ones at the end. We then use methods like cofactor expansion to find the determinant value.

Cofactor Expansion

Cofactor expansion is a method for determining the determinant of a square matrix. This technique is particularly useful for larger matrices, such as 3x3 matrices, which is often the case when checking for the collinearity of three points.

• To perform cofactor expansion, you choose a row or column from the matrix. It's sometimes easier to select the one with the most zeros to simplify calculations.
• For each element of the row or column, multiply the element by the determinant of its corresponding minor matrix, which is the matrix formed by eliminating the element's row and column.
• Additionally, apply a checkerboard pattern of plus and minus signs to each term, starting with a plus in the top left corner.

This allows us to break down the 3x3 determinant into smaller 2x2 determinants, which are easier to compute.

Matrix Representation

When dealing with a problem of collinear points, a 3x3 matrix is often used for representation. In this matrix, each row represents a point, with its x and y values occupying the first two columns. The third column is filled with ones. This formation allows us to apply techniques like cofactor expansion to check for collinearity.
The matrix representation allows us to capture all necessary information about the points in a compact form. This makes it easier to apply mathematical methods and verify relationships between points, like determining if they lie on the same line.

2x2 Matrix Determinant

For a 2x2 matrix, the determinant is calculated using a simple formula. Consider a matrix in the form:\[ \begin{array}{cc} a & b \ c & d \end{array} \] The determinant is found using the formula \( ad - bc \).
This straightforward calculation comes into play when performing cofactor expansion on a 3x3 matrix, as each element of the selected row or column leads to a 2x2 minor matrix. By solving these minor determinants, we move a step closer to finding the main determinant of the original matrix. Understanding the 2x2 matrix determinant is critical for effectively using cofactor expansion to determine potential collinearity.