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?

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.