Problem 36 - Calculus Early Transcendentals: Pearson New International Edition

Question

Show that the triangle in the plane with vertices \(\left(x_{1}, y_{1}\right),\left(x_{2}, y_{2}\right)\), and \(\left(x_{3}, y_{3}\right)\) has area equal to one-half the absolute value of the determinant $$ \left|\begin{array}{lll} x_{1} & y_{1} & 1 \\ x_{2} & y_{2} & 1 \\ x_{3} & y_{3} & 1 \end{array}\right| $$

Step-by-Step Solution

Answer

The triangle's area is half the absolute value of the determinant of its vertices matrix.

Step 1: Recall the Determinant of a 3x3 Matrix

To calculate the determinant of a 3x3 matrix given by \( A = \begin{pmatrix} a & b & c \ d & e & f \ g & h & i \end{pmatrix} \), the formula is:\[\det(A) = a(ei - fh) - b(di - fg) + c(dh - eg) \] Use this formula to find the determinant of the given matrix with vertices as rows.

Step 2: Set Up the Matrix for the Triangle

Construct the matrix using the triangle's vertices:\[\begin{pmatrix} x_1 & y_1 & 1 \ x_2 & y_2 & 1 \ x_3 & y_3 & 1 \end{pmatrix}\]This matrix is a representation of the triangle's vertices with an additional column of ones.

Step 3: Calculate the Determinant of the Matrix

Apply the determinant formula for a 3x3 matrix:\[\det(A) = x_1(y_2 \cdot 1 - y_3 \cdot 1) - y_1(x_2 \cdot 1 - x_3 \cdot 1) + 1(x_2y_3 - x_3y_2)\]Simplify each term:\[= x_1(y_2 - y_3) - y_1(x_2 - x_3) + (x_2y_3 - x_3y_2)\]

Step 4: Express the Area of the Triangle Using its Vertices

The area of a triangle given vertices \((x_1, y_1), (x_2, y_2), (x_3, y_3)\) can be found by:\[\text{Area} = \frac{1}{2} |x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2)|\]This formula results from applying the formula for the area of a polygon and is equivalent to one-half the absolute value of the determinant calculated in Step 3.

Step 5: Relate the Determinant and the Area Formula

Notice that the simplified determinant expression from Step 3:\[x_1(y_2 - y_3) - y_1(x_2 - x_3) + (x_2y_3 - x_3y_2)\]can be rewritten using the area formula:\[\det(A) = x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2)\]Thus, the area of the triangle is given by:\[\text{Area} = \frac{1}{2} | \det(A) |\]

Step 6: Final Step: Conclude the Proof

Therefore, the area of the triangle with vertices \((x_1, y_1), (x_2, y_2), (x_3, y_3)\) is equal to half of the absolute value of the determinant of the given matrix:\[\text{Area} = \frac{1}{2}\left|\begin{array}{lll} x_{1} & y_{1} & 1 \ x_{2} & y_{2} & 1 \ x_{3} & y_{3} & 1 \end{array}\right| \]

Key Concepts

Determinant of a Matrix

Determinants serve as a fundamental concept in linear algebra, providing a unique characteristic of square matrices. They help in understanding various mathematical properties, including solving linear systems, finding inverse matrices, and more.

In the context of geometry, determinants are used to calculate areas and volumes. Specifically, for a 3x3 matrix, the determinant involves expanding along any row or column to find a single numerical value.

• The formula for the determinant of a 3x3 matrix, given by:

\[A = \begin{pmatrix} a & b & c \ d & e & f \ g & h & i \end{pmatrix}\]

• is:

\[\det(A) = a(ei - fh) - b(di - fg) + c(dh - eg) \]

This formula helps to evaluate the determinant precisely by handling products and differences of elements, offering a gateway to further geometric applications.

Geometry in the Plane

Understanding geometry in the plane involves uncovering relationships between different geometric shapes, such as triangles, circles, and polygons. A fundamental property of planar geometry is calculating areas and distances between points.

One application of geometry is in finding the area of a triangle given its vertices in a coordinate plane. In this context, a determinant offers an efficient means to compute such areas directly from the coordinates of the vertices. By setting up a matrix using these vertex coordinates, we can derive the triangle's area in relation to the matrix’s determinant.

• Vertices are placed into a 3x3 matrix where:

\[\begin{pmatrix} x_1 & y_1 & 1 \ x_2 & y_2 & 1 \ x_3 & y_3 & 1 \end{pmatrix}\]

Each row represents a vertex in the coordinate plane, and computing the determinant of this matrix provides insight into geometric properties such as area.

3x3 Matrix Determinant

The process of finding a determinant for a 3x3 matrix is essential, especially when dealing with geometric problems such as finding the area of a triangle. This process involves selecting a row or column for expansion, accompanied by calculating cofactors.

Using known formulas, the determinant is calculated by recognizing the effects of each element paired with subsequent elements of adjacent rows and columns.

• It is expressed as:

\[\det(A) = x_1(y_2 \cdot 1 - y_3 \cdot 1) - y_1(x_2 \cdot 1 - x_3 \cdot 1) + 1(x_2y_3 - x_3y_2)\]

This breakdown amplifies how the values from the geometric representation translate into a determinant value, linking it to real-plane properties like a triangle's area.

Absolute Value of Determinant

The absolute value of a determinant is crucial when it comes to expressing certain geometric properties, such as area, to ensure non-negative values.

When using determinants in the context of geometry, the absolute value helps by addressing orientation issues caused by the signed nature of determinants.

• For finding the area of a triangle:

\[\text{Area} = \frac{1}{2} |\det(A)|\]

This formula signifies that regardless of whether the vertices are clockwise or counterclockwise, the area remains positive, as areas physically cannot be negative. Thus, the absolute value effectively provides a measure of the spatial size, independent of vertex ordering.