If four points `A(overline(a)), B(overline(b)), C(overline(c)) and D(overline(d))` are coplanar, then `[[overline(a), overline(b), overline(d)]]+[[overline(b), overline(c), overline(d)]]+[[overline(c), overline(a), overline(d)]]=`

Options

A. `[[overline(a), overline(c), overline(d)]]`
B. `[[overline(a), overline(b), overline(c)]]`
C. `[[overline(c), overline(b), overline(d)]]`
D. `[[overline(d), overline(b), overline(c)]]`

Answer

To solve the problem, we need to show that if four points \( A \), \( B \), \( C \), and \( D \) are coplanar, then the sum of the scalar triple products \( [[\overline{a}, \overline{b}, \overline{d}]] + [[\overline{b}, \overline{c}, \overline{d}]] + [[\overline{c}, \overline{a}, \overline{d}]] = 0 \).

### Step-by-Step Solution:

1. **Understanding Coplanarity**:
Since the points \( A \), \( B \), \( C \), and \( D \) are coplanar, it means that they lie in the same plane. For vectors, this implies that the scalar triple product of any three vectors formed by these points will be zero.

2. **Scalar Triple Product**:
The scalar triple product \( [[\overline{u}, \overline{v}, \overline{w}]] \) can be calculated using the formula:
\[ [[\overline{u}, \overline{v}, \overline{w}]] = \overline{u} \cdot (\overline{v} \times \overline{w}) \]
This represents the volume of the parallelepiped formed by the vectors \( \overline{u} \), \( \overline{v} \), and \( \overline{w} \). If the volume is zero, it indicates that the vectors are coplanar.

3. **Calculating Each Term**:
- For the first term \( [[\overline{a}, \overline{b}, \overline{d}]] \):
\[ [[\overline{a}, \overline{b}, \overline{d}]] = \overline{a} \cdot (\overline{b} \times \overline{d}) = 0 \]
because \( A, B, D \) are coplanar.

- For the second term \( [[\overline{b}, \overline{c}, \overline{d}]] \):
\[ [[\overline{b}, \overline{c}, \overline{d}]] = \overline{b} \cdot (\overline{c} \times \overline{d}) = 0 \]
because \( B, C, D \) are coplanar.

- For the third term \( [[\overline{c}, \overline{a}, \overline{d}]] \):
\[ [[\overline{c}, \overline{a}, \overline{d}]] = \overline{c} \cdot (\overline{a} \times \overline{d}) = 0 \]
because \( C, A, D \) are coplanar.

4. **Summing the Terms**:
Now we sum all three scalar triple products:
\[ [[\overline{a}, \overline{b}, \overline{d}]] + [[\overline{b}, \overline{c}, \overline{d}]] + [[\overline{c}, \overline{a}, \overline{d}]] = 0 + 0 + 0 = 0 \]

5. **Conclusion**:
Therefore, we conclude that:
\[ [[\overline{a}, \overline{b}, \overline{d}]] + [[\overline{b}, \overline{c}, \overline{d}]] + [[\overline{c}, \overline{a}, \overline{d}]] = 0 \]

### Final Answer:
The answer is \( 0 \).