ABCDEF is a regular hexagon with centre a the origin such that `vec(AB)+vec(EB)+vec(FC)= lamda vec(ED) then lamda =` (A) 2 (B) 4 (C) 6 (D) 3

Options

A. `-4`
B. 4
C. `-2`
D. 2

Answer

To solve the problem, we need to find the value of \( \lambda \) in the equation:

\[ \vec{AB} + \vec{ED} + \vec{FC} = \lambda \vec{ED} \]

Given that \( ABCDEF \) is a regular hexagon centered at the origin, we can start by analyzing the vectors involved.

### Step 1: Understand the properties of the regular hexagon
In a regular hexagon, all sides are equal, and the angles between adjacent sides are \( 120^\circ \). The vectors representing the sides can be expressed in terms of a common length \( x \).

### Step 2: Define the vectors
Let:
- \( \vec{AB} = \vec{ED} = \vec{FO} = \vec{OC} = x \)

### Step 3: Find \( \vec{FC} \)
The vector \( \vec{FC} \) can be expressed as the sum of \( \vec{FO} \) and \( \vec{OC} \):
\[ \vec{FC} = \vec{FO} + \vec{OC} = x + x = 2x \]

### Step 4: Substitute the vectors into the equation
Now we can substitute the values of the vectors into the original equation:
\[ \vec{AB} + \vec{ED} + \vec{FC} = x + x + 2x = 4x \]

### Step 5: Set up the equation with \( \lambda \)
We know that:
\[ \lambda \vec{ED} = \lambda x \]
Thus, we can equate the two sides:
\[ 4x = \lambda x \]

### Step 6: Solve for \( \lambda \)
Assuming \( x \neq 0 \), we can divide both sides by \( x \):
\[ 4 = \lambda \]
So, the value of \( \lambda \) is:
\[ \lambda = 4 \]

### Conclusion
The answer is \( \lambda = 4 \), which corresponds to option (B).

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