Consider the regular hexagon ABCDEF with centre at O (origin).
Q. Five forces AB,AC,AD,AE,AF act at the vertex A of a regular hexagon ABCDEF. Then, their resultant is (a)3AO (b)2AO (c)4AO (d)6AO

Options

A. 3AO
B. 2AO
C. 4AO
D. 6AO

Explanation

To solve the problem of finding the resultant of the five forces acting at vertex A of a regular hexagon ABCDEF, we can follow these steps:

### Step 1: Understand the Forces
We have five forces acting at point A, which are directed towards the vertices of the hexagon:
- Force AB (towards B)
- Force AC (towards C)
- Force AD (towards D)
- Force AE (towards E)
- Force AF (towards F)

### Step 2: Represent the Forces as Vectors
We can represent these forces as vectors. Assuming the side length of the hexagon is 1 unit, we can express the position vectors of points B, C, D, E, and F in terms of their angles from the origin O (which is the center of the hexagon):
- \( \vec{B} = \vec{A} + \vec{AB} \)
- \( \vec{C} = \vec{A} + \vec{AC} \)
- \( \vec{D} = \vec{A} + \vec{AD} \)
- \( \vec{E} = \vec{A} + \vec{AE} \)
- \( \vec{F} = \vec{A} + \vec{AF} \)

### Step 3: Calculate the Resultant Force
The resultant force \( \vec{R} \) can be calculated by summing these vectors:
\[ \vec{R} = \vec{AB} + \vec{AC} + \vec{AD} + \vec{AE} + \vec{AF} \]

### Step 4: Use Symmetry of the Hexagon
Due to the symmetry of the regular hexagon, the sum of the position vectors of points B, C, D, E, and F will equal zero when considered from the center O. Thus:
\[ \vec{B} + \vec{C} + \vec{D} + \vec{E} + \vec{F} = 0 \]
This means that:
\[ \vec{R} = \vec{AB} + \vec{AC} + \vec{AD} + \vec{AE} + \vec{AF} = 5\vec{A} - \vec{O} \]

### Step 5: Express the Resultant in Terms of OA
Since \( \vec{O} \) is the origin, we can express the resultant force in terms of the vector \( \vec{OA} \):
\[ \vec{R} = 6\vec{OA} \]

### Conclusion
Thus, the resultant of the five forces acting at vertex A of the regular hexagon is:
\[ \vec{R} = 6\vec{OA} \]
The correct answer is (d) \( 6AO \).

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