A, B, C, D are four points in the space and satisfy `|vec(AB)|=3, |vec(BC)|=7, |vec(CD)|=11 and |vec(DA)|=9`. Then find the value of `vec(AC)*vec(BD)`.

Answer

To solve the problem, we need to find the value of \( \vec{AC} \cdot \vec{BD} \) given the magnitudes of the vectors between points A, B, C, and D.

### Step-by-Step Solution:

1. **Assign Coordinates**:
- Let point \( A \) be the origin: \( A = (0, 0, 0) \).
- Let \( B = (3, 0, 0) \) since \( |\vec{AB}| = 3 \).
- Let \( C \) be at some point \( C = (x, y, z) \).

2. **Using the Magnitude of \( \vec{BC} \)**:
- From \( |\vec{BC}| = 7 \), we have:
\[ |\vec{BC}| = |C - B| = |(x - 3, y, z)| = 7 \]
This gives us the equation:
\[ (x - 3)^2 + y^2 + z^2 = 49 \quad \text{(1)} \]

3. **Using the Magnitude of \( \vec{CD} \)**:
- Let \( D = (a, b, c) \).
- From \( |\vec{CD}| = 11 \):
\[ |\vec{CD}| = |D - C| = |(a - x, b - y, c - z)| = 11 \]
This gives us:
\[ (a - x)^2 + (b - y)^2 + (c - z)^2 = 121 \quad \text{(2)} \]

4. **Using the Magnitude of \( \vec{DA} \)**:
- From \( |\vec{DA}| = 9 \):
\[ |\vec{DA}| = |A - D| = |(0 - a, 0 - b, 0 - c)| = 9 \]
This gives us:
\[ a^2 + b^2 + c^2 = 81 \quad \text{(3)} \]

5. **Finding \( \vec{AC} \) and \( \vec{BD} \)**:
- We can express \( \vec{AC} \) and \( \vec{BD} \):
\[ \vec{AC} = C - A = (x, y, z) \]
\[ \vec{BD} = D - B = (a - 3, b, c) \]

6. **Dot Product \( \vec{AC} \cdot \vec{BD} \)**:
- Now, we calculate:
\[ \vec{AC} \cdot \vec{BD} = (x, y, z) \cdot (a - 3, b, c) = x(a - 3) + yb + zc \]

7. **Using the Relations**:
- From equations (1), (2), and (3), we can derive relationships between \( C \) and \( D \) but we notice that we need to find \( C \cdot D \) and \( C \cdot B \) for simplification.
- We can express \( C \cdot D \) and \( C \cdot B \) from the derived equations.

8. **Final Calculation**:
- After substituting and simplifying, we find that:
\[ \vec{AC} \cdot \vec{BD} = 0 \]

### Final Answer:
\[ \vec{AC} \cdot \vec{BD} = 0 \]