https://file.aiquickdraw.com/mathbot/file/1727005923888kwmyyx10.jpg

löse mir die aufgabe 9 mit den koordinaten: H(1/2/2), E(3/2/2), F(3/6/2), G(1/6/2), C(1/6/-1), D(1/2/-1), A(3/2/-1), B(3/6/-1), H(1/2/2).

To solve **task 9**, we are asked to determine the coordinates of the points \( M_1, M_2, M_3, M_4 \), which are the intersection points of the diagonals of various planes in the cuboid, as well as the coordinates of the vectors \( \overrightarrow{M_1M_2}, \overrightarrow{M_2M_3}, \overrightarrow{M_3M_4}, \overrightarrow{M_4M_1} \).

### Step 1: Coordinates of \( M_1, M_2, M_3, M_4 \)

We will find the coordinates of each \( M_i \) by determining the midpoints of the diagonals in the specified planes.

- **\( M_1 \)**: Midpoint of the diagonal of rectangle \( ABCD \)
  \[
  M_1 = \left( \frac{x_A + x_C}{2}, \frac{y_A + y_C}{2}, \frac{z_A + z_C}{2} \right)
  = \left( \frac{3+1}{2}, \frac{2+6}{2}, \frac{-1 + (-1)}{2} \right)
  = (2, 4, -1)
  \]

- **\( M_2 \)**: Midpoint of the diagonal of rectangle \( BCGF \)
  \[
  M_2 = \left( \frac{x_B + x_G}{2}, \frac{y_B + y_G}{2}, \frac{z_B + z_G}{2} \right)
  = \left( \frac{3+1}{2}, \frac{6+6}{2}, \frac{-1+2}{2} \right)
  = (2, 6, 0.5)
  \]

- **\( M_3 \)**: Midpoint of the diagonal of rectangle \( CDHG \)
  \[
  M_3 = \left( \frac{x_C + x_H}{2}, \frac{y_C + y_H}{2}, \frac{z_C + z_H}{2} \right)
  = \left( \frac{1+1}{2}, \frac{6+2}{2}, \frac{-1+2}{2} \right)
  = (1, 4, 0.5)
  \]

- **\( M_4 \)**: Midpoint of the diagonal of rectangle \( ADHE \)
  \[
  M_4 = \left( \frac{x_A + x_E}{2}, \frac{y_A + y_E}{2}, \frac{z_A + z_E}{2} \right)
  = \left( \frac{3+3}{2}, \frac{2+2}{2}, \frac{-1+2}{2} \right)
  = (3, 2, 0.5)
  \]

### Step 2: Coordinates of the Vectors

Now, we find the coordinates of the vectors by subtracting the coordinates of the points involved.

- **\( \overrightarrow{M_1M_2} \)**
  \[
  \overrightarrow{M_1M_2} = M_2 - M_1 = (2, 6, 0.5) - (2, 4, -1) = (0, 2, 1.5)
  \]

- **\( \overrightarrow{M_2M_3} \)**
  \[
  \overrightarrow{M_2M_3} = M_3 - M_2 = (1, 4, 0.5) - (2, 6, 0.5) = (-1, -2, 0)
  \]

- **\( \overrightarrow{M_3M_4} \)**
  \[
  \overrightarrow{M_3M_4} = M_4 - M_3 = (3, 2, 0.5) - (1, 4, 0.5) = (2, -2, 0)
  \]

- **\( \overrightarrow{M_4M_1} \)**
  \[
  \overrightarrow{M_4M_1} = M_1 - M_4 = (2, 4, -1) - (3, 2, 0.5) = (-1, 2, -1.5)
  \]

### Summary:

- Coordinates of \( M_1 = (2, 4, -1) \)
- Coordinates of \( M_2 = (2, 6, 0.5) \)
- Coordinates of \( M_3 = (1, 4, 0.5) \)
- Coordinates of \( M_4 = (3, 2, 0.5) \)

- Vector \( \overrightarrow{M_1M_2} = (0, 2, 1.5) \)
- Vector \( \overrightarrow{M_2M_3} = (-1, -2, 0) \)
- Vector \( \overrightarrow{M_3M_4} = (2, -2, 0) \)
- Vector \( \overrightarrow{M_4M_1} = (-1, 2, -1.5) \)

Do you have any further questions or need clarification?

### Related Questions:
1. How do you find the midpoint of a diagonal in a 3D space?
2. How do you calculate the vector between two points in 3D?
3. What is the significance of diagonal midpoints in a cuboid?
4. Can you explain the vector subtraction method for any two points?
5. How would you calculate the distance between two midpoints?

### Tip:
When calculating vectors between points, always subtract coordinates in the same order (final minus initial) to ensure the correct direction.

Solve problem 9 using the coordinates: H(1/2/2), E(3/2/2), F(3/6/2), G(1/6/2), C(1/6/-1), D(1/2/-1), A(3/2/-1), B(3/6/-1), H(1/2/2).

This detailed solution explains how to find the midpoints of diagonals and calculate vectors in a 3D cuboid, using the coordinates H(1/2/2), E(3/2/2), F(3/6/2), G(1/6/2), C(1/6/-1), D(1/2/-1), A(3/2/-1), and B(3/6/-1). It includes step-by-step explanations on using the midpoint formula and vector subtraction.

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