If ABCDEF is a regualr hexagon, then `vec(AC) + vec(AD) + vec(EA) + vec(FA)=`

Options

A. `2 vec(AB)`
B. `3 vec (AB)`
C. `vec(AB)`
D. `vec0`

Answer

To solve the problem, we need to find the value of the expression \( \vec{AC} + \vec{AD} + \vec{EA} + \vec{FA} \) for a regular hexagon ABCDEF. Let's break it down step by step.

### Step 1: Define the Vectors
We start by defining the vectors for the sides of the hexagon. Let:
- \( \vec{AB} = \vec{a} \)
- \( \vec{BC} = \vec{b} \)

In a regular hexagon, all sides are equal, and the angles between adjacent sides are \( 120^\circ \).

### Step 2: Calculate \( \vec{AC} \)
The vector \( \vec{AC} \) can be expressed as:
\[ \vec{AC} = \vec{AB} + \vec{BC} = \vec{a} + \vec{b} \]

### Step 3: Calculate \( \vec{AD} \)
In a regular hexagon, the vector \( \vec{AD} \) can be calculated as:
\[ \vec{AD} = 2\vec{b} \]
This is because \( \vec{AD} \) spans two sides of the hexagon.

### Step 4: Calculate \( \vec{EA} \)
The vector \( \vec{EA} \) can be expressed as:
\[ \vec{EA} = -\vec{b} - \vec{b} = -2\vec{b} \]
This is because \( \vec{EA} \) goes in the opposite direction of \( \vec{BC} \) and \( \vec{CD} \).

### Step 5: Calculate \( \vec{FA} \)
The vector \( \vec{FA} \) can be expressed as:
\[ \vec{FA} = -(\vec{b} - \vec{a}) = \vec{a} - \vec{b} \]
This is because \( \vec{FA} \) is in the opposite direction of \( \vec{CD} \).

### Step 6: Substitute Values into the Expression
Now we substitute the values we found into the expression:
\[ \vec{AC} + \vec{AD} + \vec{EA} + \vec{FA} = (\vec{a} + \vec{b}) + (2\vec{b}) + (-2\vec{b}) + (\vec{a} - \vec{b}) \]

### Step 7: Simplify the Expression
Now we simplify the expression:
\[ = \vec{a} + \vec{b} + 2\vec{b} - 2\vec{b} + \vec{a} - \vec{b} \]
Combining like terms:
\[ = 2\vec{a} + 0\vec{b} = 2\vec{a} \]

### Final Answer
Thus, the final result is:
\[ \vec{AC} + \vec{AD} + \vec{EA} + \vec{FA} = 2\vec{a} \]