If A, B, C, D be any four points and E and F be the middle points of AC and BD respectively, then `A vec(B) + C vec(B) +C vec (D) + vec(AD)` is equal to

Options

A. `3 vec(EF)`
B. `4vec(EF)`
C. `4 vec(FE)`
D. `3 vec(FE)`

Explanation

To solve the problem, we need to find the expression \( \vec{AB} + \vec{CB} + \vec{CD} + \vec{AD} \) in terms of the midpoints \( E \) and \( F \).

### Step 1: Define the Position Vectors
Let the position vectors of points \( A, B, C, D \) be represented as:
- \( \vec{A} \)
- \( \vec{B} \)
- \( \vec{C} \)
- \( \vec{D} \)

### Step 2: Find the Midpoints
The midpoints \( E \) and \( F \) can be defined as:
- \( E \) is the midpoint of \( AC \):
\[ \vec{E} = \frac{1}{2} (\vec{A} + \vec{C}) \]
- \( F \) is the midpoint of \( BD \):
\[ \vec{F} = \frac{1}{2} (\vec{B} + \vec{D}) \]

### Step 3: Express the Vectors
Now we express the vectors \( \vec{AB}, \vec{CB}, \vec{CD}, \vec{AD} \):
- \( \vec{AB} = \vec{B} - \vec{A} \)
- \( \vec{CB} = \vec{B} - \vec{C} \)
- \( \vec{CD} = \vec{D} - \vec{C} \)
- \( \vec{AD} = \vec{D} - \vec{A} \)

### Step 4: Combine the Vectors
Now, we combine these vectors:
\[ \vec{AB} + \vec{CB} + \vec{CD} + \vec{AD} = (\vec{B} - \vec{A}) + (\vec{B} - \vec{C}) + (\vec{D} - \vec{C}) + (\vec{D} - \vec{A}) \]

### Step 5: Simplify the Expression
Combining like terms:
\[ = 2\vec{B} + 2\vec{D} - 2\vec{A} - 2\vec{C} \]
Factoring out the common factor of 2:
\[ = 2(\vec{B} + \vec{D} - \vec{A} - \vec{C}) \]

### Step 6: Relate to Midpoints
Now, we can relate this to the midpoints \( E \) and \( F \):
From the definition of \( E \) and \( F \):
\[ \vec{E} = \frac{1}{2}(\vec{A} + \vec{C}) \quad \text{and} \quad \vec{F} = \frac{1}{2}(\vec{B} + \vec{D}) \]
Thus, we can express:
\[ \vec{B} + \vec{D} - \vec{A} - \vec{C} = 2(\vec{F} - \vec{E}) \]

### Final Result
Substituting this back into our expression:
\[ \vec{AB} + \vec{CB} + \vec{CD} + \vec{AD} = 2 \cdot 2(\vec{F} - \vec{E}) = 4(\vec{F} - \vec{E}) \]

Therefore, the final result is:
\[ \vec{AB} + \vec{CB} + \vec{CD} + \vec{AD} = 4 \vec{EF} \]