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

The triangle ABC is a right triangle, with a right angle at vertex B.

Step 1: Calculate vector AB

To check if the angle at point B is 90 degrees, we first need to calculate the vectors AB and BC. The vector \( \overrightarrow{AB} \) can be found by subtracting the coordinates of point A from point B. Thus, \( \overrightarrow{AB} = B - A = (3 - 6, 1 - 3, -1 - 3) = (-3, -2, -4) \).

Step 2: Calculate vector BC

Next, we calculate the vector BC by subtracting the coordinates of point B from point C. Thus, \( \overrightarrow{BC} = C - B = (-1 - 3, 10 - 1, -2.5 + 1) = (-4, 9, -1.5) \).

Step 3: Calculate the dot product of AB and BC

The dot product of two vectors \( \vec{u} = (u_1, u_2, u_3) \) and \( \vec{v} = (v_1, v_2, v_3) \) is calculated as \( \vec{u} \cdot \vec{v} = u_1v_1 + u_2v_2 + u_3v_3 \). For vectors \( \overrightarrow{AB} = (-3, -2, -4) \) and \( \overrightarrow{BC} = (-4, 9, -1.5) \), the dot product is:\[ (-3)\times(-4) + (-2)\times9 + (-4)\times(-1.5) = 12 - 18 + 6 = 0 \]

Step 4: Conclusion about a right angle

Since the dot product of \( \overrightarrow{AB} \) and \( \overrightarrow{BC} \) is zero, the angle between these two vectors is 90 degrees. Therefore, triangle \( \triangle ABC \) has a right angle at vertex B, confirming that it is a right triangle.

Key Concepts

Vector Calculations

Vectors are fundamental in many fields of science and engineering. In the context of a right triangle, understanding vectors helps us grasp directions and distances in a coordinate system. A vector essentially represents a quantity that has both magnitude and direction. When dealing with vectors for geometry, such as finding the sides of a triangle, we often perform vector calculations like subtraction or addition.
For example, to calculate vector \( \overrightarrow{AB} \) for a triangle with points A and B, we'd subtract the coordinates of A from those of B. This operation, \( (B_x - A_x, B_y - A_y, B_z - A_z) \), gives us another vector that points from A to B. In our scenario, \( \overrightarrow{AB} = (-3, -2, -4) \).
Some basic operations you can perform with vectors include:

• **Addition:** Adding corresponding components of vectors.
• **Subtraction:** Subtracting corresponding components to find a new vector's direction.
• **Scalar Multiplication:** Multiplying each component by a scalar for direction scaling.

These operations form the core of vector arithmetic, which is crucial for understanding geometric relationships in spaces.

Dot Product

The dot product, also known as the scalar product, is a powerful tool to determine the angle between two vectors. It's calculated by multiplying corresponding components of two vectors and then summing those products. The formula for the dot product of vectors \( \vec{u} = (u_1, u_2, u_3) \) and \( \vec{v} = (v_1, v_2, v_3) \) is \( \vec{u} \cdot \vec{v} = u_1v_1 + u_2v_2 + u_3v_3 \).
When the dot product of two vectors is zero, it indicates that the vectors are perpendicular, or in simpler terms, they form a 90-degree angle. This concept was used to prove that the triangle given in our problem is right-angled.
By calculating the dot product of vectors \( \overrightarrow{AB} = (-3, -2, -4) \) and \( \overrightarrow{BC} = (-4, 9, -1.5) \), we determined:
\[ (-3) \times (-4) + (-2) \times 9 + (-4) \times (-1.5) = 0 \]
The result of zero confirms the angle between the vectors is indeed 90 degrees, meaning there's a right angle at vertex B, verifying the triangle is a right triangle.

3D Geometry

3D geometry involves understanding shapes and figures in a three-dimensional space, defined by three axes (x, y, and z). This makes it a more complex and richly detailed area compared to two-dimensional geometry.
In our exercise, we consider a triangle with coordinates in 3D space. When working in 3D, the additional dimension offers greater depth, allowing us to analyze angles and distances more comprehensively, but also requiring careful calculation.
Key points to remember in 3D geometry include:

• **Coordinates:** Points are expressed as (x, y, z), each representing position on the respective axis.
• **Distance calculation:** Uses the distance formula: \( \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2 + (z_2-z_1)^2} \).
• **Vectors in 3D:** Similar to 2D but includes an additional z-component for describing direction and magnitude.

In this domain, we've shown how to confirm shapes like right triangles by employing vector calculations and understanding spatial relationships in the third dimension. It's a fascinating study area with applications ranging from physics to architecture.