Let A₁ A₂ A₃ A₄ A₅ A₆ A₁ represent a regular hexagon, as shown in the figure. The six sides, taken in sequence, represent six vectors. By utilizing the x-components of these vectors and the fact that their resultant is zero, determine the value of the expression 0 + 3 + 2 3 + 3 3 + 4 3 + 5 3 .

Step-by-step solution

Let the side length of the regular hexagon be a . The six sides of the hexagon taken in sequence can be represented as six vectors: A₁ A₂ , A₂ A₃ , A₃ A₄ , A₄ A₅ , A₅ A₆ , and A₆ A₁ . Since these vectors form a closed polygon, their vector sum is zero: A₁ A₂ + A₂ A₃ + A₃ A₄ + A₄ A₅ + A₅ A₆ + A₆ A₁ = 0 Consequently, the sum of their components along any axis must also be zero. Let us consider the components along the X-axis. The angle made by each vector with the positive X-axis is as follows: A₁ A₂ is along the X-a