If `a.(b xx c) = 3` then

Options

A. `c. (a xx b) =-3`
B. `a.(c xx b) =-3`
C. `b.(a xxc) =3`
D. `(a xx c), b =3`

Answer

To solve the problem where \( a \cdot (b \times c) = 3 \), we need to analyze the expression and apply the properties of vector operations, particularly focusing on the scalar triple product.

### Step-by-step Solution:

1. **Understanding the Expression**:
The expression \( a \cdot (b \times c) \) represents the scalar triple product of vectors \( a, b, \) and \( c \). This product gives the volume of the parallelepiped formed by these vectors.

2. **Using the Property of Scalar Triple Product**:
The scalar triple product has a property that allows us to rearrange the vectors. Specifically, we know that:
\[ a \cdot (b \times c) = b \cdot (c \times a) = c \cdot (a \times b) \]
This means we can express \( a \cdot (b \times c) \) in terms of other combinations of the vectors.

3. **Applying the Rearrangement**:
From the property mentioned, we can write:
\[ a \cdot (b \times c) = 3 \implies b \cdot (c \times a) = 3 \quad \text{and} \quad c \cdot (a \times b) = 3 \]
This indicates that the scalar triple product remains constant regardless of the order of the vectors.

4. **Interpreting the Result**:
Since \( a \cdot (b \times c) = 3 \), we can conclude that the volume of the parallelepiped formed by vectors \( a, b, \) and \( c \) is 3. This means that the vectors are not coplanar and indeed form a three-dimensional shape.

5. **Finding the Correct Option**:
If we were given multiple-choice options, we would look for an option that correctly represents one of the equivalent forms of the scalar triple product. The correct option would involve \( a, b, \) and \( c \) in a way that maintains the equality of the scalar triple product.

### Final Answer:
The correct interpretation of the expression \( a \cdot (b \times c) = 3 \) leads us to conclude that the volume of the parallelepiped formed by these vectors is 3, and we can express this in various equivalent forms based on the properties of the scalar triple product.