If `a*b=b*c=c*a=0," then [abc]` is equal to

Options

A. 0
B. 1
C. -1
D. `abs(a)abs(b)abs(c)`

Answer

To solve the problem, we need to find the value of the scalar triple product \([abc]\) given that \(a \cdot b = b \cdot c = c \cdot a = 0\). This means that the vectors \(a\), \(b\), and \(c\) are mutually orthogonal to each other.

### Step-by-Step Solution:

1. **Understanding the Scalar Triple Product**:
The scalar triple product \([abc]\) can be represented as the determinant of the matrix formed by the vectors \(a\), \(b\), and \(c\):
\[ [abc] = a \cdot (b \times c) \]
This represents the volume of the parallelepiped formed by the vectors \(a\), \(b\), and \(c\).

2. **Using the Given Information**:
Since \(a \cdot b = 0\), \(b \cdot c = 0\), and \(c \cdot a = 0\), it indicates that the vectors are orthogonal to each other. Therefore, we can conclude that:
- \(a\) is perpendicular to \(b\)
- \(b\) is perpendicular to \(c\)
- \(c\) is perpendicular to \(a\)

3. **Calculating the Scalar Triple Product**:
The scalar triple product can also be expressed as:
\[ [abc] = |a| |b| |c| \cos \theta \]
where \(\theta\) is the angle between the vectors. Since \(a\), \(b\), and \(c\) are mutually orthogonal, the angles between each pair of vectors are \(90^\circ\). Therefore, \(\cos 90^\circ = 0\).

4. **Conclusion**:
Since the cosine of the angle between each pair of vectors is zero, we have:
\[ [abc] = |a| |b| |c| \cdot 0 = 0 \]
Thus, the value of the scalar triple product \([abc]\) is:
\[ \boxed{0} \]