Vector Analysis in 2D Polygons

00:00 10 marks

Consider a convex polygon where each side is represented by a vector. This problem explores vector sums in polygonal structures and their decomposition.

Part a

7 minutes 4 marks

Question

Given a convex pentagon (5 sides), with consecutive side vectors \(\vec{A}, \vec{B}, \vec{C}, \vec{D}\), and \(\vec{E}\), demonstrate that \(\vec{A} + \vec{B} + \vec{C} + \vec{D} + \vec{E} = \vec{0}\). Explain why this vector sum being zero is a necessary condition for the polygon to be closed.

Hint Hint

Hint

Consider the path traced by the vectors around the polygon. What must be true if you return to the starting point?

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Complete Answer

For a closed polygon, the sum of the vectors representing its sides, taken in order, must be \(\vec{0}\). This is because moving along each vector consecutively brings you back to the starting point, resulting in zero net displacement. For a pentagon, this is \(\vec{A} + \vec{B} + \vec{C} + \vec{D} + \vec{E} = \vec{0}\).

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Part b

8 minutes 6 marks

Question

Extend this concept to a general convex n-sided polygon. Provide a formal proof using vector components to show that the sum of its consecutive side vectors is always \(\vec{0}\). Explain how decomposing vectors into horizontal and vertical components reinforces the understanding of polygon closure.

Hint Hint

Hint

Represent each vector in terms of its horizontal and vertical components. Consider the sum of all horizontal components and the sum of all vertical components separately to show they each sum to zero. This uses the principle that if the total displacement in both x and y directions is zero, the overall displacement is zero.

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Complete Answer

Let \(\vec{V_1}, \vec{V_2}, …, \vec{V_n}\) be the consecutive side vectors of a convex n-sided polygon. Each vector can be decomposed into horizontal and vertical components: \(\vec{V_i} = V_{ix}\,\hat{i} + V_{iy}\,\hat{j}\). For the polygon to be closed, the total displacement in both the horizontal and vertical directions must be zero. Thus, \(\sum_{i=1}^{n} V_{ix} = 0\) and \(\sum_{i=1}^{n} V_{iy} = 0\). Therefore, the sum of all vectors is \(\sum_{i=1}^{n} \vec{V_i} = (\sum_{i=1}^{n} V_{ix})\,\hat{i} + (\sum_{i=1}^{n} V_{iy})\,\hat{j} = 0\,\hat{i} + 0\,\hat{j} = \vec{0}\). Decomposing vectors shows that closure in 2D space requires closure in both independent directions, ensuring no net displacement in either direction. This component approach simplifies the proof by reducing a 2D vector problem into two independent 1D scalar problems.

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