If `vecr.veca=vecr.vecb=vecr.vecc=0` for some non-zero vectro `vecr`, then the value of `[(veca, vecb, vecc)]` is

Options

A. `2`
B. `3`
C. `0`
D. none of these

Explanation

To solve the problem, we need to analyze the given conditions and determine the value of the scalar triple product \([ \vec{a}, \vec{b}, \vec{c} ]\).

### Step-by-Step Solution:

1. **Understanding the Given Conditions**:
We are given that:
\[ \vec{r} \cdot \vec{a} = 0, \quad \vec{r} \cdot \vec{b} = 0, \quad \vec{r} \cdot \vec{c} = 0 \]
This means that the vector \(\vec{r}\) is orthogonal (perpendicular) to the vectors \(\vec{a}\), \(\vec{b}\), and \(\vec{c}\).

2. **Implications of Orthogonality**:
Since \(\vec{r}\) is orthogonal to \(\vec{a}\), \(\vec{b}\), and \(\vec{c}\), it implies that all three vectors \(\vec{a}\), \(\vec{b}\), and \(\vec{c}\) lie in the same plane that is perpendicular to \(\vec{r}\). Therefore, \(\vec{a}\), \(\vec{b}\), and \(\vec{c}\) are coplanar.

3. **Scalar Triple Product**:
The scalar triple product \([ \vec{a}, \vec{b}, \vec{c} ]\) is defined as:
\[ [ \vec{a}, \vec{b}, \vec{c} ] = \vec{a} \cdot (\vec{b} \times \vec{c}) \]
If the vectors \(\vec{a}\), \(\vec{b}\), and \(\vec{c}\) are coplanar, then the cross product \(\vec{b} \times \vec{c}\) will yield a vector that is perpendicular to the plane formed by \(\vec{b}\) and \(\vec{c}\). Since \(\vec{a}\) lies in the same plane, the dot product \(\vec{a} \cdot (\vec{b} \times \vec{c})\) will be zero.

4. **Conclusion**:
Therefore, the value of the scalar triple product \([ \vec{a}, \vec{b}, \vec{c} ]\) is:
\[ [ \vec{a}, \vec{b}, \vec{c} ] = 0 \]

### Final Answer:
\[ [ \vec{a}, \vec{b}, \vec{c} ] = 0 \]