Problem 20 - College Algebra and Calculus: An Applied Approach

Question

Use a determinant to determine whether the points are collinear. $$ (1,7),(0,4),(-1,2) $$

Step-by-Step Solution

Answer

The points (1,7), (0,4), and (-1,2) are collinear.

Step 1: Setup the 3x3 Matrix

Set up a 3x3 matrix for the three points (1,7), (0,4), and (-1,2). The first row will be composed of all 1s, the second row will contain the x-coordinates (1,0, and -1), and the third row will be the y-coordinates (7,4, and 2). So, the matrix is \[ \begin{bmatrix} 1 & 1 & 1 \\ 1 & 0 & -1 \\ 7 & 4 & 2 \end{bmatrix} \]

Step 2: Calculate the determinant

Calculate the determinant of the matrix. The determinant of a 3x3 matrix can be calculated by using the rule of Sarrus. Following that rule, we obtain that the determinant, det, is\[ \text{det} = (1*0*2) + (1*4*1) + (1*1*-1) - (1*0*7) - (1*1*2) - (1*4*-1) = 0. \]

Step 3: Analyze the Result

Since the determinant is zero, the points (1,7), (0,4), and (-1,2) are collinear.

Key Concepts

Collinearity

Points are collinear when they lie on the same straight line. In geometry, one way to determine if three points are collinear is by using the concept of determinants, which involves assessing a matrix composed of these points. If the determinant of this matrix is zero, the points are collinear. This means that they do not create any area, highlighting that they lie on the same line. Understanding and verifying collinearity can be useful in various mathematical contexts, such as coordinate geometry and vector analysis.

3x3 Matrix

A 3x3 matrix is an array of numbers organized into three rows and three columns. It's often used in problems involving determinants, especially when dealing with points in a plane. For the problem of collinearity, a 3x3 matrix is set up as follows:

• The first row consists of all ones, which acts as a placeholder to facilitate the calculation of the determinant.
• The second row includes the x-coordinates of the points.
• The third row comprises the y-coordinates of the points.

Creating this matrix correctly is crucial for accurately determining whether the points are collinear.

Calculation of Determinants

The determinant of a 3x3 matrix is a special number calculated from its elements. In this context, it's used to determine if points are collinear. The calculation provides a scalar value that encapsulates information about the matrix:

• If the determinant is zero, it implies that the rows (or columns) are linearly dependent, indicating collinearity in the context of points.
• A non-zero determinant suggests that the points are not collinear.

Understanding how to compute and interpret the determinant is essential for solving many mathematical problems.

Sarrus' Rule

Sarrus' rule is a method to easily calculate the determinant of a 3x3 matrix. It involves a specific pattern:

• Multiply the main diagonal (upper left to lower right) elements and add them together.
• Cross-multiply elements above and below the main diagonal, adding these products.
• Subtract the products calculated from diagonal lines that run from top-right to bottom-left.

Using Sarrus' rule can simplify complex determinant calculations, making it an invaluable tool in matrices involving 3-point collinearity checks.