Problem 18 - Calculus Early Transcendentals: Pearson New International Edition

Question

Show that the triangle \(A B C\) is a right triangle if the vertices are \(A(6,3,3), B(3,1,-1)\), and \(C(-1,10,-2.5) .\) Hint: Check the angle at \(B\).

Step-by-Step Solution

Answer

Triangle \(ABC\) is a right triangle at vertex \(B\) because \( \vec{AB} \cdot \vec{BC} = 0 \).

Step 1: Find Vector AB

To find vector \( \vec{AB} \), subtract the coordinates of point \( A \) from the coordinates of point \( B \). Thus, \( \vec{AB} = B - A = (3-6, 1-3, -1-3) = (-3, -2, -4) \).

Step 2: Find Vector BC

To find vector \( \vec{BC} \), subtract the coordinates of point \( B \) from the coordinates of point \( C \). Thus, \( \vec{BC} = C - B = (-1-3, 10-1, -2.5+1) = (-4, 9, -1.5) \).

Step 3: Calculate Dot Product of Vectors AB and BC

The dot product of two vectors \( \vec{AB} \) and \( \vec{BC} \) is given by \( \vec{AB} \cdot \vec{BC} = (-3)(-4) + (-2)(9) + (-4)(-1.5) \). Simplifying gives: \( 12 - 18 + 6 = 0 \).

Step 4: Conclude Based on Dot Product

Since the dot product \( \vec{AB} \cdot \vec{BC} = 0 \), this indicates that the angle between the vectors is 90 degrees. Therefore, triangle \( ABC \) is a right triangle at vertex \( B \).

Key Concepts

Understanding the Dot Product

The dot product is an essential operation in vector mathematics, especially relevant to problems involving triangles and geometry. It provides a way to determine the angle between two vectors.

In the context of right triangles, the dot product becomes particularly powerful. If the dot product of two vectors is zero, the vectors are perpendicular.

• For vectors \( \vec{u} = (x_1, y_1, z_1) \) and \( \vec{v} = (x_2, y_2, z_2) \), the dot product is calculated as:

\[ \vec{u} \cdot \vec{v} = x_1 x_2 + y_1 y_2 + z_1 z_2 \]
In our example, we used the vectors \( \vec{AB} = (-3, -2, -4) \) and \( \vec{BC} = (-4, 9, -1.5) \). By calculating \( \vec{AB} \cdot \vec{BC} \) and obtaining 0, we confirmed that the angle between vectors \( \vec{AB} \) and \( \vec{BC} \) was 90 degrees, ensuring triangle ABC is a right triangle at vertex B.

The Process of Vector Subtraction

Vector subtraction is a simple but critical skill in analytic geometry. It allows us to determine the direction and magnitude of one line segment relative to another.

When subtracting vectors to find a new vector, the coordinates of the initial point are subtracted from the coordinates of the terminal point. For example, to find vector \( \vec{AB} \), subtract the coordinates of point A from point B:

• Point A: \( (6, 3, 3) \)
• Point B: \( (3, 1, -1) \)

This yields \( \vec{AB} = B - A = (3-6, 1-3, -1-3) = (-3, -2, -4) \). Similarly, vector \( \vec{BC} \) was found by subtracting B from C, giving \( \vec{BC} = (-4, 9, -1.5) \).

Vector subtraction is often used to find the relative positions and movements of points in 3D space, laying the groundwork for calculating other properties, such as the dot product.

Applying Analytic Geometry

Analytic geometry, or coordinate geometry, bridges algebra with geometry. By using coordinates in a plane or space, you can explore geometric problems with algebraic methods. When determining the nature of a triangle such as triangle ABC, analytic geometry offers tools like vector operations and the dot product.

To decide whether triangle ABC was a right triangle, we used analytic geometry to:

• Determine vectors \( \vec{AB} \) and \( \vec{BC} \)
• Calculate their dot product

This approach simplifies complex geometry into algebraic operations without the need for protractors or direct angle measurements. By ensuring the dot product is zero, analytic geometry confirms that the triangle’s angle at vertex B is 90 degrees, proving it’s a right triangle.

Such exercises highlight how analytic geometry provides clarity and precision in solving geometric problems using algebraic techniques.