Problem 5 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use a determinant to find the area of the figure with the given vertices. (-2,4),(2,3),(-1,5)

Step-by-Step Solution

Answer

The area of the triangle can be calculated by taking half of the absolute value of the determinant of the matrix formed by the coordinates of the vertices, this in turn represents the area of the given polygon.

Step 1: Put coordinates in a matrix

The vertices are given as (-2,4), (2,3), and (-1,5). Put these coordinates into a matrix form. Each coordinate in a row and add 1 at the end of each row as a third column. Let's call this matrix A. Matrix A will look like: \[A = \begin{bmatrix} -2 & 4 & 1 \ 2 & 3 & 1 \ -1 & 5 & 1 \end{bmatrix}\]

Step 2: Finding the determinant of the matrix

Next, find the determinant of the matrix A. Remember, you can apply any of the methods to find the determinant such as minors and cofactors, diagonals method, Sarrus' rule. Also, determinant of 3x3 matrix can take negative values, depending on the orientation of points in space.

Step 3: Calculating the area

The area of the triangle is half of the absolute value of the determinant of the matrix A. In mathematical terms, \(Area = \frac{1}{2} \times |Determinant(A)|\)

Key Concepts

Matrix Representation of Coordinates

In geometry, we often deal with coordinates that help to define points in a two-dimensional plane. These coordinates are typically lists of numbers, but when we're trying to use mathematical operations on them, putting them into a matrix form can be very useful. A matrix is basically a way to organize numbers into a grid of rows and columns.
For the exercise given, we have vertices represented by the coordinates

• (-2, 4)
• (2, 3)
• (-1, 5)

To turn a set of coordinates into a matrix, we write each coordinate as a row. It's also common to add an additional column of 1s, which plays a crucial role in calculations involving determinants. This format is handy when dealing with geometric transformations and calculations. Our matrix with these coordinates becomes a 3x3 matrix. Each row corresponds to a point, and the additional column with 1s helps with certain geometric interpretations and calculations.
By putting our coordinates into a matrix, it allows us to use matrix operations, such as calculating a determinant, which is very helpful for finding areas of geometric shapes, among other things.

Area Calculation Using Determinants

Determinants provide an elegant method for calculating the area of geometric figures, such as triangles, directly from their coordinates. When vertices are placed into a matrix, the determinant of this matrix holds significant information. For a triangle, the formula to find the area using determinants is:\[ Area = \frac{1}{2} \times |Determinant(A)| \]This formula arises because the determinant gives twice the signed area of the triangle. Therefore, dividing it by 2 and taking the absolute value ensures we get the correct magnitude for the area, disregarding any orientation or ordering of the vertices used.
In this problem, after putting the coordinates into a matrix and finding the determinant, you can apply this formula.
Calculating the determinant involves several steps, such as using methods including minors and cofactors, or Sarrus’ rule if applicable for a 3x3 matrix. Once you have determined this value, the above formula allows you to derive the exact area effectively.

Properties of Determinants

Determinants have fascinating properties that are crucial in various geometric and algebraic problems. Some of the key properties of determinants include:

• Linear Transformation: The determinant of a square matrix helps to understand how linear transformations scale areas or volumes. A zero determinant suggests the figure collapses into a lower dimension.
• Sign of Determinant: Indicates the orientation or "handedness" of the basis. A positive determinant means oriented in one direction, while a negative means reversed.
• Multiplicative Property: If you have two matrices, A and B, the determinant of their product equals the product of their determinants, i.e., \( \text{Det}(A \cdot B) = \text{Det}(A) \times \text{Det}(B) \).
• Row and Column Operations: Swapping two rows or columns changes the sign of the determinant. Multiplying a row by a constant multiplies the determinant by that constant. Adding or subtracting rows doesn't change the determinant.

Understanding these properties helps grasp the role of determinants beyond calculating areas. They are pivotal in solutions of linear equations, eigenvalue problems, and in understanding spaces' dimensional transformations.