Problem 75 - Algebra and Trigonometry

Question

Vectors That Form a Polygon Suppose that \(n\) vectors can be placed head to tail in the plane so that they form a polygon. (The figure shows the case of a hexagon.) Explain why the sum of these vectors is 0 .

Step-by-Step Solution

Answer

The sum is 0 because the vectors form a closed loop, returning to the start.

Step 1: Understanding Vector Addition and Polygon Formation

When vectors are placed 'head to tail', it means that the tail of each vector starts where the head of the previous vector ends. Forming a polygon with vectors implies that the final position of the last vector coincides with the starting position of the first vector, effectively completing a closed loop.

Step 2: Establishing the Sum of Vectors

Consider placing the vectors in sequence: \( \mathbf{v}_1, \mathbf{v}_2, \ldots, \mathbf{v}_n \). When these vectors form a polygon, the endpoint of \( \mathbf{v}_n \) aligns with the starting point of \( \mathbf{v}_1 \), returning to the initial position. Mathematically, this means the sum of these vectors must return the displacement to zero.

Step 3: Mathematical Representation

Since forming a polygon brings you back to the starting point, the sum of all displacements must be zero: \[ \mathbf{v}_1 + \mathbf{v}_2 + \cdots + \mathbf{v}_n = \mathbf{0} \]. This equation represents that the total movement described by these vectors cancels out, leading to no net displacement.

Key Concepts

Polygon Formation

When we talk about polygon formation using vectors, we consider a scenario where vectors are aligned head-to-tail. This means the end of one vector seamlessly connects with the start of the next. Imagine drawing in this manner where you begin with one vector, and then another starts where the first ended. Continue this until you use all your vectors.

A polygon is created when the last vector you've added comes back to the original starting point. Visualize this as building shapes like a triangle, square, or hexagon with your vectors as the sides. For the figure to close and become a polygon, each vector must follow in the sequence from the end of the previous, forming a continuous and closed shape. Here are some key points about polygon formation:

• The vectors must connect head-to-tail.
• The sequence should end back where it started.
• The path forms a recognizable shape (polygon).

By completing this loop, you create a geometric figure that visually reinforces the concept at the heart of this exercise: the resultant sum of vectors that forms a polygon is zero.

Zero Vector

The zero vector is a unique and important concept in vector mathematics. It represents a vector that has no magnitude and no direction. When vectors arranged in a polygon result in the zero vector, it indicates that all movements or displacements described by these vectors cancel each other out.

In the context of polygon formation, when the journey of vectors head-to-tail returns to the starting point, we effectively have no net travel from our original position. This cumulation coming to zero means that the sum of all vectors is a zero vector. Here are a few characteristics to remember:

• The zero vector is denoted as \( \mathbf{0} \).
• It implies no net movement or displacement.
• The sum of vectors that form a closed shape results in the zero vector.

Appreciating this outcome helps us understand that all the individual directional movements of the vectors intersect precisely to have no overall effect, bringing us symbolically back to the same point in space.

Closed Loop

A closed loop is a complete circuit that effectively returns to the starting point. In vector terms, it means the sum of all vectors in the set results in the zero vector, indicating no change in position from where you started. This concept is central to understanding why polygons formed by vectors lead to a sum of zero.

Consider each vector as a step or a movement. If each step leads you back to the initial position, then overall, you have not moved—this is the closed loop in action. Here are some aspects of a closed loop regarding vectors:

• All vectors head-to-tail should bring you back to the origin.
• The loop physically symbolizes no total displacement.
• It creates a complete shape without any breaks or ends.

Understanding closed loops is crucial for grasping why polygons are significant in vector theory; they showcase the principle that multiple movements can balance out to zero when arranged in a specific, closed sequence.