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.

Step 1: Understand the problem

We need to prove that the area of a triangle with given vertices is half the absolute value of a specific determinant.

Step 2: Converse determinant formula to matrix form

Write the determinant for the coordinates of the triangle vertices, which is given as: \[\left|\begin{array}{ccc}x_1 & y_1 & 1 \x_2 & y_2 & 1 \x_3 & y_3 & 1\end{array}\right|\]

Step 3: Expression for triangle area

Recall that the area of a triangle with vertices \((x_1, y_1), (x_2, y_2), (x_3, y_3)\) is given by the formula: \[ \text{Area} = \frac{1}{2} \left| x_1(y_2-y_3) + x_2(y_3-y_1) + x_3(y_1-y_2) \right| \]

Step 4: Express determinant in terms of area

Calculate the determinant: \[\left|\begin{array}{ccc}x_1 & y_1 & 1 \x_2 & y_2 & 1 \x_3 & y_3 & 1\end{array}\right|= x_1(y_2-y_3) + x_2(y_3-y_1) + x_3(y_1-y_2)\]

Step 5: Compare both expressions

The expression obtained from the determinant in the previous step matches the expression inside the absolute value in the area formula. Therefore, the area of the triangle is \[\frac{1}{2} \left| \left|\begin{array}{ccc}x_1 & y_1 & 1 \x_2 & y_2 & 1 \x_3 & y_3 & 1\end{array}\right| \right|\]

Step 6: Conclusion

Thus, it is proven that the area of the triangle is equal to one-half the absolute value of the determinant.

Key Concepts

Determinant Formula

The determinant formula is a powerful tool in mathematics that helps us compute the area of a triangle formed by three points in a coordinate plane. This formula can be particularly useful because it provides a straightforward method for performing calculations that might otherwise be cumbersome. The determinant formula for the area of a triangle with vertices at \( (x_1, y_1), (x_2, y_2), (x_3, y_3) \) is given by the equation: \[ \text{Area} = \frac{1}{2} \left| x_1(y_2-y_3) + x_2(y_3-y_1) + x_3(y_1-y_2) \right| \] This formula is derived from the geometric properties of a triangle and leverages the benefits of linear algebra to simplify calculations with the determinant. By utilizing the concept of a determinant, we can represent the vertices in a matrix form, leading to a quick and effective computation of the area without needing to graph the triangle or measure any angles.

Matrix Representation

Matrix representation in the context of finding the area of a triangle means that we can express the information about the triangle's vertices in a matrix form. This matrix is a 3x3 matrix constructed from the coordinates of the vertices. Here's how you put it together: - Each row represents one vertex - Each row consists of the x-coordinate, y-coordinate, and an auxiliary number 1 You arrange the vertices as: \[ \left|\begin{array}{ccc} x_1 & y_1 & 1 \ x_2 & y_2 & 1 \ x_3 & y_3 & 1 \end{array}\right| \] This matrix configuration is crucial because it simplifies the process of calculating determinants, which in turn helps in computing the triangle's area. The matrix is constructed such that applying the determinant formula translates directly into a calculation of the relevant expression, encapsulating much of the complexity into a simple format. This method is highly beneficial in mathematical problem-solving as it provides a clear, consistent means of organizing and visualizing the vertex information.

Coordinates of Vertices

The coordinates of vertices in a triangle are essential pieces of information required to determine the area using the determinant formula. In any given triangle, each vertex has a specific position in the coordinate plane described by an ordered pair \( (x, y) \). These coordinates are crucial because they define the triangle's precise shape and its location. Here is how you can imagine this process: - Each vertex is a point where two sides of the triangle meet. - The coordinates \((x_1, y_1), (x_2, y_2), (x_3, y_3)\) describe the exact location of each vertex. Understanding the role of these coordinates helps underline why they are entered into a matrix and subsequently worked out through determinants. They not only define the edges of the triangle but also directly influence the calculation of its area. In a sense, this area formula is a direct extrapolation from their differences and interrelations, encapsulated within the determinant.

Mathematical Proof

The mathematical proof for showing that the area of a triangle is half the absolute value of its determinant involves both geometric intuition and algebraic manipulation. The key to this proof is demonstrating that the formula \[ \text{Area} = \frac{1}{2} \left| \Delta \right| \] where \( \Delta \) is the determinant of the 3x3 matrix representing the triangle's vertices, holds true. To accomplish this, we start by calculating the determinant of the matrix: \[ \Delta = \left|\begin{array}{ccc} x_1 & y_1 & 1 \ x_2 & y_2 & 1 \ x_3 & y_3 & 1 \end{array}\right| = x_1(y_2-y_3) + x_2(y_3-y_1) + x_3(y_1-y_2) \] This expression for \( \Delta \) perfectly aligns with traditional algebraic formulas for computing the area of a triangle from three known vertices. Both expressions inside the absolute value sign confirm the geometric interpretation of the area when visualized on the coordinate plane, proving that they indeed represent the same quantity. By being succinct yet comprehensive, this proof not only verifies correctness but also shows how linear algebra techniques connect to geometric understanding.