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Виріши задачу яка на фото

The problem in the image is about vectors in a parallelogram \( OABC \). The vectors given are:

\[
\overrightarrow{OA} = \vec{a}, \quad \overrightarrow{OB} = \vec{b}
\]

You are asked to express the vectors \( \overrightarrow{MO}, \overrightarrow{MA}, \overrightarrow{MB}, \overrightarrow{MC} \) in terms of \( \vec{a} \) and \( \vec{b} \), where \( M \) is the intersection point of the diagonals of the parallelogram.

### Approach:

In a parallelogram, the diagonals bisect each other. So, the point \( M \), being the midpoint of both diagonals, divides each diagonal into two equal parts.

1. The diagonal \( AC \) is represented by \( \overrightarrow{AC} \). We can express \( C \) using the parallelogram rule, where:
   \[
   \overrightarrow{OC} = \overrightarrow{OA} + \overrightarrow{OB} = \vec{a} + \vec{b}
   \]

2. The midpoint \( M \) of the diagonals will have the position vector:
   \[
   \overrightarrow{OM} = \frac{1}{2} \overrightarrow{OC} = \frac{1}{2} (\vec{a} + \vec{b})
   \]

Now, we can express the required vectors:

1. **\( \overrightarrow{MO} \)**:
   \[
   \overrightarrow{MO} = -\overrightarrow{OM} = -\frac{1}{2} (\vec{a} + \vec{b})
   \]

2. **\( \overrightarrow{MA} \)**:
   \[
   \overrightarrow{MA} = \overrightarrow{OA} - \overrightarrow{OM} = \vec{a} - \frac{1}{2} (\vec{a} + \vec{b}) = \frac{1}{2} \vec{a} - \frac{1}{2} \vec{b}
   \]

3. **\( \overrightarrow{MB} \)**:
   \[
   \overrightarrow{MB} = \overrightarrow{OB} - \overrightarrow{OM} = \vec{b} - \frac{1}{2} (\vec{a} + \vec{b}) = \frac{1}{2} \vec{b} - \frac{1}{2} \vec{a}
   \]

4. **\( \overrightarrow{MC} \)**:
   \[
   \overrightarrow{MC} = \overrightarrow{OC} - \overrightarrow{OM} = (\vec{a} + \vec{b}) - \frac{1}{2} (\vec{a} + \vec{b}) = \frac{1}{2} (\vec{a} + \vec{b})
   \]

### Final Results:
- \( \overrightarrow{MO} = -\frac{1}{2} (\vec{a} + \vec{b}) \)
- \( \overrightarrow{MA} = \frac{1}{2} \vec{a} - \frac{1}{2} \vec{b} \)
- \( \overrightarrow{MB} = \frac{1}{2} \vec{b} - \frac{1}{2} \vec{a} \)
- \( \overrightarrow{MC} = \frac{1}{2} (\vec{a} + \vec{b}) \)

Would you like further details on any specific part?

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**Related Questions:**
1. How can you prove that diagonals of a parallelogram bisect each other?
2. What happens if \( \vec{a} \) and \( \vec{b} \) are orthogonal vectors?
3. How do you find the area of a parallelogram using vectors?
4. What is the physical interpretation of the vector \( \overrightarrow{OM} \)?
5. Can this method be applied to any convex quadrilateral?

**Tip:** In vector problems involving geometry, identifying relationships like midpoints and bisected diagonals can simplify the calculations considerably.

Explore how to express vectors at the midpoint of a parallelogram using vector addition laws and geometry. This advanced tutorial is perfect for university-level students and covers key concepts such as vector addition, subtraction, and the properties of parallelogram diagonals.

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