The problem asks for the vector coordinates of $\overrightarrow{AC} + \overrightarrow{BD}$, where the coordinates of the points are:

• $A(1, 0, 1)$
• $B(2, -1, 1)$
• $C(-2, -2, 1)$
• $D(1, -2, 1)$

Step 1: Find the vector $\overrightarrow{AC}$

The vector $\overrightarrow{AC}$ is obtained by subtracting the coordinates of $A$ from those of $C$:

$\overrightarrow{AC} = C - A = (-2 - 1, -2 - 0, 1 - 1) = (-3, -2, 0)$

Step 2: Find the vector $\overrightarrow{BD}$

Similarly, the vector $\overrightarrow{BD}$ is obtained by subtracting the coordinates of $B$ from those of $D$:

$\overrightarrow{BD} = D - B = (1 - 2, -2 - (-1), 1 - 1) = (-1, -1, 0)$

Step 3: Add the vectors $\overrightarrow{AC}$ and $\overrightarrow{BD}$

Now, we add the two vectors:

$\overrightarrow{AC} + \overrightarrow{BD} = (-3, -2, 0) + (-1, -1, 0) = (-4, -3, 0)$

Thus, the vector coordinates of $\overrightarrow{AC} + \overrightarrow{BD}$ are $(-4, -3, 0)$.

Would you like further details or explanations on any part of this solution?

Here are 5 related questions to further your understanding:

1. How do you subtract two vectors in three-dimensional space?
2. What is the significance of a vector's direction and magnitude?
3. How can you find the length (magnitude) of a vector like $\overrightarrow{AC}$?
4. How does vector addition work geometrically?
5. Can vectors be added if they are not in the same coordinate space?

Tip: When adding vectors, make sure to add the corresponding components (x with x, y with y, and z with z) to avoid errors.