If `overline(a) and overline(b)` are parallel vectors `[[overline(a), overline(b), overline(c)]]`=

Options

A. `0`
B. `1`
C. `2`
D. `4`

Answer

To solve the problem, we need to determine the value of the triple product of vectors \( \overline{a}, \overline{b}, \overline{c} \) given that vectors \( \overline{a} \) and \( \overline{b} \) are parallel.

### Step-by-step Solution:

1. **Understanding Parallel Vectors**:
- Vectors \( \overline{a} \) and \( \overline{b} \) are parallel, which means they are scalar multiples of each other. This can be expressed as:
\[ \overline{b} = k \overline{a} \]
for some scalar \( k \).

2. **Cross Product of Parallel Vectors**:
- The cross product of two parallel vectors is always zero. Therefore, we can write:
\[ \overline{a} \times \overline{b} = 0 \]

3. **Triple Product Definition**:
- The triple product of three vectors \( \overline{a}, \overline{b}, \overline{c} \) is defined as:
\[ \overline{a} \cdot (\overline{b} \times \overline{c}) \]
- Alternatively, it can also be expressed as:
\[ \overline{a} \times \overline{b} \cdot \overline{c} \]

4. **Substituting the Cross Product**:
- Since we established that \( \overline{a} \times \overline{b} = 0 \), we can substitute this into the triple product:
\[ \overline{a} \cdot (\overline{b} \times \overline{c}) = \overline{a} \cdot 0 = 0 \]

5. **Conclusion**:
- Therefore, the value of the triple product \( [\overline{a}, \overline{b}, \overline{c}] \) is:
\[ 0 \]

### Final Answer:
The triple product \( [\overline{a}, \overline{b}, \overline{c}] = 0 \).