If diagonals of a parallelogram ABCD intersect each other in M, then `bar(OA) + bar(OB) + bar(OC) + bar(OD)` =

Options

A. `bar(OM)`
B. 2`bar(OM)`
C. 3`bar(OM)`
D. 4`bar(OM)`

Answer

To solve the problem, we need to find the expression for the sum of the position vectors of the vertices of the parallelogram ABCD, given that the diagonals intersect at point M.

### Step-by-Step Solution:

1. **Understanding the Position Vectors**:
Let the position vectors of points A, B, C, and D be denoted as \(\vec{A}\), \(\vec{B}\), \(\vec{C}\), and \(\vec{D}\) respectively. The diagonals of the parallelogram intersect at point M.

2. **Using the Properties of Parallelograms**:
In a parallelogram, the diagonals bisect each other. Therefore, the position vector of point M can be expressed as:
\[ \vec{M} = \frac{\vec{A} + \vec{C}}{2} = \frac{\vec{B} + \vec{D}}{2} \]

3. **Expressing the Sum of Position Vectors**:
We need to calculate \(\vec{OA} + \vec{OB} + \vec{OC} + \vec{OD}\). The position vector of the origin O is denoted as \(\vec{O}\). Thus, we can express:
\[ \vec{OA} = \vec{A} - \vec{O}, \quad \vec{OB} = \vec{B} - \vec{O}, \quad \vec{OC} = \vec{C} - \vec{O}, \quad \vec{OD} = \vec{D} - \vec{O} \]

4. **Combining the Vectors**:
Now, we can combine these vectors:
\[ \vec{OA} + \vec{OB} + \vec{OC} + \vec{OD} = (\vec{A} - \vec{O}) + (\vec{B} - \vec{O}) + (\vec{C} - \vec{O}) + (\vec{D} - \vec{O}) \]
Simplifying this, we get:
\[ = \vec{A} + \vec{B} + \vec{C} + \vec{D} - 4\vec{O} \]

5. **Using the Diagonal Properties**:
Since \(\vec{M} = \frac{\vec{A} + \vec{C}}{2}\) and \(\vec{M} = \frac{\vec{B} + \vec{D}}{2}\), we can express \(\vec{A} + \vec{C}\) and \(\vec{B} + \vec{D}\) as:
\[ \vec{A} + \vec{C} = 2\vec{M}, \quad \vec{B} + \vec{D} = 2\vec{M} \]
Therefore, we can substitute:
\[ \vec{A} + \vec{B} + \vec{C} + \vec{D} = (\vec{A} + \vec{C}) + (\vec{B} + \vec{D}) = 2\vec{M} + 2\vec{M} = 4\vec{M} \]

6. **Final Expression**:
Substituting this back into our earlier equation:
\[ \vec{OA} + \vec{OB} + \vec{OC} + \vec{OD} = 4\vec{M} - 4\vec{O} = 4(\vec{M} - \vec{O}) \]

### Conclusion:
Thus, the final result is:
\[ \vec{OA} + \vec{OB} + \vec{OC} + \vec{OD} = 4(\vec{M} - \vec{O}) \]