https://file.aiquickdraw.com/mathbot/file/17264946399174jhcmrnm.jpg

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.

Find the vector coordinates of \(\overrightarrow{AC} + \overrightarrow{BD}\) where the points are A(1, 0, 1), B(2, -1, 1), C(-2, -2, 1), and D(1, -2, 1).

Learn how to find the vector coordinates of \(\overrightarrow{AC} + \overrightarrow{BD}\) with points A(1, 0, 1), B(2, -1, 1), C(-2, -2, 1), and D(1, -2, 1). This problem covers vector addition and subtraction in 3D geometry, suitable for Grades 9-12.

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