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 $ABCD$ = \left( \frac{3+1}{2}, \frac{2+6}{2}, \frac{-1 + (-1)}{2} \right) = (2, 4, -1)$$
• $M_2$: Midpoint of the diagonal of rectangle $BCGF$ = \left( \frac{3+1}{2}, \frac{6+6}{2}, \frac{-1+2}{2} \right) = (2, 6, 0.5)$$
• $M_3$: Midpoint of the diagonal of rectangle $CDHG$ = \left( \frac{1+1}{2}, \frac{6+2}{2}, \frac{-1+2}{2} \right) = (1, 4, 0.5)$$
• $M_4$: Midpoint of the diagonal of rectangle $ADHE$ = \left( \frac{3+3}{2}, \frac{2+2}{2}, \frac{-1+2}{2} \right) = (3, 2, 0.5)$$

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.