For non-coplanar vectors a, b and c, `abs((a times b)*c)=abs(a) abs(b) abs(c)` holds if and only if

Options

A. `a*b=b*c=c*a=0`
B. `a*b=0=b*c`
C. `a*b=0=c*a`
D. `b*c=0=c*a`

Answer

To solve the problem, we need to determine the condition under which the equation \( |\mathbf{(a \times b) \cdot c}| = |\mathbf{a}| |\mathbf{b}| |\mathbf{c}| \) holds true for non-coplanar vectors \(\mathbf{a}\), \(\mathbf{b}\), and \(\mathbf{c}\).

### Step-by-Step Solution:

1. **Understanding the Scalar Triple Product**:
The expression \( |\mathbf{(a \times b) \cdot c}| \) represents the absolute value of the scalar triple product of vectors \(\mathbf{a}\), \(\mathbf{b}\), and \(\mathbf{c}\). This scalar triple product can be interpreted geometrically as the volume of the parallelepiped formed by the three vectors.

2. **Using the Property of Scalar Triple Product**:
The scalar triple product can be expressed in terms of determinants:
\[ |\mathbf{(a \times b) \cdot c}| = |\det(\mathbf{a}, \mathbf{b}, \mathbf{c})| \]
This determinant can also be represented as:
\[ \det(\mathbf{a}, \mathbf{b}, \mathbf{c}) = |\mathbf{a}| |\mathbf{b}| |\mathbf{c}| \cos(\theta) \]
where \(\theta\) is the angle between the vector \(\mathbf{c}\) and the vector formed by \(\mathbf{a} \times \mathbf{b}\).

3. **Setting Up the Condition**:
The equation we want to satisfy is:
\[ |\det(\mathbf{a}, \mathbf{b}, \mathbf{c})| = |\mathbf{a}| |\mathbf{b}| |\mathbf{c}| \]
This implies:
\[ |\det(\mathbf{a}, \mathbf{b}, \mathbf{c})| = |\mathbf{a}| |\mathbf{b}| |\mathbf{c}| \cos(\theta) \]

4. **Analyzing the Condition**:
For the above equality to hold, we need:
\[ \cos(\theta) = 1 \quad \text{or} \quad \theta = 0 \]
This means that the vector \(\mathbf{c}\) must be parallel to the vector \(\mathbf{a} \times \mathbf{b}\).

5. **Conclusion**:
Therefore, the condition under which the equation \( |\mathbf{(a \times b) \cdot c}| = |\mathbf{a}| |\mathbf{b}| |\mathbf{c}| \) holds true is when \(\mathbf{c}\) is perpendicular to the plane formed by \(\mathbf{a}\) and \(\mathbf{b}\), which can also be stated as:
\[ \mathbf{a} \cdot \mathbf{b} = 0, \quad \mathbf{b} \cdot \mathbf{c} = 0, \quad \mathbf{c} \cdot \mathbf{a} = 0 \]
This implies that the vectors \(\mathbf{a}\), \(\mathbf{b}\), and \(\mathbf{c}\) are mutually orthogonal.