If `barr.bara=barr.barb=barr.barc=0` for non-zero vector `barr` then the value of `(bara barb barc)` is

Options

A. 2
B. 3
C. 0
D. None of these

Answer

To solve the problem, we need to find the value of the scalar triple product \( ( \bar{a} \, \bar{b} \, \bar{c} ) \) given that \( \bar{r} \cdot \bar{a} = \bar{r} \cdot \bar{b} = \bar{r} \cdot \bar{c} = 0 \) for a non-zero vector \( \bar{r} \).

### Step-by-Step Solution:

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

**Hint**: Recall that if a vector is orthogonal to another vector, their dot product is zero.

2. **Using the Scalar Triple Product**:
The scalar triple product \( ( \bar{a} \, \bar{b} \, \bar{c} ) \) can be expressed in terms of a dot product and a cross product:
\[ ( \bar{a} \, \bar{b} \, \bar{c} ) = \bar{r} \cdot ( \bar{a} \times \bar{b} ) \]
This means we can express the scalar triple product as the dot product of \( \bar{r} \) with the cross product of \( \bar{a} \) and \( \bar{b} \).

**Hint**: Remember that the scalar triple product gives the volume of the parallelepiped formed by the three vectors.

3. **Substituting the Conditions**:
Since \( \bar{r} \cdot \bar{a} = 0 \), \( \bar{r} \cdot \bar{b} = 0 \), and \( \bar{r} \cdot \bar{c} = 0 \), we can conclude that:
\[ \bar{r} \cdot ( \bar{a} \times \bar{b} ) = 0 \]
This indicates that the vector \( \bar{r} \) is orthogonal to the vector \( \bar{a} \times \bar{b} \).

**Hint**: A vector is orthogonal to another vector if their dot product is zero.

4. **Conclusion**:
Since \( \bar{r} \) is orthogonal to the plane formed by \( \bar{a} \) and \( \bar{b} \), it implies that the volume of the parallelepiped formed by \( \bar{a} \), \( \bar{b} \), and \( \bar{c} \) is zero. Therefore, we conclude that:
\[ ( \bar{a} \, \bar{b} \, \bar{c} ) = 0 \]

**Final Answer**: The value of \( ( \bar{a} \, \bar{b} \, \bar{c} ) \) is \( 0 \).