Problem 55 - Thomas Calculus

Question

Suppose that \(A, B\) , and \(C\) are vertices of a triangle and that \(a, b,\) and \(c\) are, respectively, the midpoints of the opposite sides. Show that \(\overrightarrow{A a}+\overrightarrow{B b}+\overrightarrow{C c}=0.\)

Step-by-Step Solution

Answer

The vector sum \( \overrightarrow{A a} + \overrightarrow{B b} + \overrightarrow{C c} = 0 \).

Step 1: Define position vectors

Let the position vectors of vertices \( A, B, \) and \( C \) be \( \overrightarrow{A}, \overrightarrow{B}, \) and \( \overrightarrow{C} \), respectively. The vectors for midpoints can be expressed in terms of these position vectors.

Step 2: Determine midpoints in vector form

The midpoint \( a \) of side \( BC \) has position vector \( \overrightarrow{a} = \frac{\overrightarrow{B} + \overrightarrow{C}}{2} \), the midpoint \( b \) of side \( AC \) has position vector \( \overrightarrow{b} = \frac{\overrightarrow{A} + \overrightarrow{C}}{2} \), and the midpoint \( c \) of side \( AB \) has position vector \( \overrightarrow{c} = \frac{\overrightarrow{A} + \overrightarrow{B}}{2} \).

Step 3: Express vectors from vertices to midpoints

The vectors from vertices to midpoints are: \(\overrightarrow{A a} = \overrightarrow{a} - \overrightarrow{A} = \frac{\overrightarrow{B} + \overrightarrow{C}}{2} - \overrightarrow{A}, \)\(\overrightarrow{B b} = \overrightarrow{b} - \overrightarrow{B} = \frac{\overrightarrow{A} + \overrightarrow{C}}{2} - \overrightarrow{B}, \) and\(\overrightarrow{C c} = \overrightarrow{c} - \overrightarrow{C} = \frac{\overrightarrow{A} + \overrightarrow{B}}{2} - \overrightarrow{C}. \)

Step 4: Find the sum of these vectors

Calculate the sum: \( \overrightarrow{A a} + \overrightarrow{B b} + \overrightarrow{C c} = \left(\frac{\overrightarrow{B} + \overrightarrow{C}}{2} - \overrightarrow{A}\right) + \left(\frac{\overrightarrow{A} + \overrightarrow{C}}{2} - \overrightarrow{B}\right) + \left(\frac{\overrightarrow{A} + \overrightarrow{B}}{2} - \overrightarrow{C}\right) \).

Step 5: Simplify the expression

Combine like terms and observe cancellation: \( \frac{\overrightarrow{B} + \overrightarrow{C} + \overrightarrow{A} + \overrightarrow{C} + \overrightarrow{A} + \overrightarrow{B}}{2} - (\overrightarrow{A} + \overrightarrow{B} + \overrightarrow{C}) \). This simplifies to:\( \frac{2(\overrightarrow{A} + \overrightarrow{B} + \overrightarrow{C})}{2} - (\overrightarrow{A} + \overrightarrow{B} + \overrightarrow{C}) = 0 \).

Key Concepts

Midpoint Formula

The midpoint formula is a crucial concept in geometry, especially when dealing with line segments. This formula allows you to find the exact center point between two locations. When applied in vector geometry, the midpoint of a line segment connecting two points \((x_1, y_1)\) and \((x_2, y_2)\) in a Cartesian plane is calculated as:

• \((\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2})\)

For position vectors, the formula works similarly. If \(\overrightarrow{A}\) and \(\overrightarrow{B}\) are vectors representing two points, the midpoint \(\overrightarrow{M}\) is determined by:

• \(\overrightarrow{M} = \frac{\overrightarrow{A} + \overrightarrow{B}}{2}\)

This concept is vital for finding line segment midpoints in vector problems. It simplifies the process of determining the middle point of any given line segment by merely averaging the respective vector components. This method can be easily extended to three dimensions as well, making it extremely versatile. Understanding and correctly applying the midpoint formula is often the key to solving complex geometry problems involving triangles, quadrilaterals, or other polygons.

Position Vectors

Position vectors are foundational concepts in vector geometry. A position vector locates a point in space relative to an origin, typically denoted \(O\). For any point \(P(x, y, z)\) in three-dimensional space, the position vector \(\overrightarrow{OP}\) is defined as:

• \(\overrightarrow{OP} = \langle x, y, z \rangle\)

This vector is crucial for expressing points in a space, finding distances, and determining directions.
Position vectors make operations like addition, subtraction, and scaling straightforward when working in vector form. In the context of the exercise, the position vectors \(\overrightarrow{A}, \overrightarrow{B}, \overrightarrow{C}\) represent the vertices of a triangle in a plane.
Using these vectors, we can find midpoints, determine lines, and perform other operations such as dot products and cross products. Understanding the basics of position vectors allows for a deeper comprehension of geometric properties and relationships within a system of points.

Triangle Geometry

Triangle geometry involves studying the properties and relations within triangles. An essential part of this study is understanding how lines and points interact over the structure of triangles. Within the realm of vector geometry, these interactions can be expressed through position vectors and vector operations.
In the exercise, the points \(A\), \(B\), and \(C\) form a triangle. Using the midpoints of the opposite sides allows us to explore the centroid concept, even though our problem specifically uses the midpoints to show zero sum of vectors. Such exercises help illustrate fundamental geometric principles that, while initially subtle, underscore powerful symmetries found in nature.
Vectors like \(\overrightarrow{Aa}\), \(\overrightarrow{Bb}\), and \(\overrightarrow{Cc}\) (expressed as differences between midpoint vectors and vertices) are essential in proving geometric relationships. Here, the sum of these vectors equals zero, reflecting the balanced nature of these geometric constructs. Understanding these concepts allows students to see the harmony in mathematical structures, aiding in problem-solving and theoretical proofs.