`[vecaxx vecb " " vecc xx vecd " " vecexx vecf]` is equal to
(a)`[veca vecb vecd] [vecc vece vecf]-[veca vecb vecc] [vecd vece vecf]`
(b)`[veca vecb vece ] [vecf vecc vecd] - [veca vecb vecf] [vece vecc vecd] `
(c)`[vecc vecd veca] [vecb vece vecf] - [veca vecd vecb] [vecavece vecf]`
(d)`[veca vecc vece] [vecb vecd vecf]`

Options

A. `[veca vecb vecd] [vecc vece vecf]-[veca vecb vecc] [vecd vece vecf]`
B. `[veca vecb vece ] [vecf vecc vecd] - [veca vecb vecf] [vece vecc vecd] `
C. `[vecc vecd veca] [vecb vece vecf] - [veca vecd vecb] [vecavece vecf]`
D. `[veca vecc vece] [vecb vecd vecf]`

Explanation

To solve the problem, we need to evaluate the expression \([ \vec{a} \times \vec{b} \; \vec{c} \times \vec{d} \; \vec{e} \times \vec{f} ]\) and determine which of the given options is equivalent to it.

### Step-by-Step Solution:

1. **Understanding the Scalar Triple Product**:
The expression \([ \vec{a} \times \vec{b} \; \vec{c} \times \vec{d} \; \vec{e} \times \vec{f} ]\) represents the scalar triple product of the vectors involved. The scalar triple product can be expressed as:
\[ \vec{a} \times \vec{b} \cdot (\vec{c} \times \vec{d} \times \vec{e} \times \vec{f}) \]

2. **Using the Properties of Cross and Dot Products**:
We can rewrite the expression using the properties of cross and dot products. The scalar triple product has the property that:
\[ \vec{u} \times (\vec{v} \times \vec{w}) = (\vec{u} \cdot \vec{w}) \vec{v} - (\vec{u} \cdot \vec{v}) \vec{w} \]
Applying this property, we can express \((\vec{c} \times \vec{d}) \times \vec{e}\) and \((\vec{c} \times \vec{d}) \times \vec{f}\).

3. **Expanding the Expression**:
We can expand the expression as follows:
\[ [\vec{a} \times \vec{b}] \cdot [(\vec{c} \times \vec{d}) \times \vec{e} \times \vec{f}] = [\vec{a} \times \vec{b}] \cdot [\vec{c} \times \vec{d} \times \vec{e} - \vec{c} \times \vec{f}] \]

4. **Identifying the Correct Option**:
Now, we will analyze the options provided:
- Option (a): \([ \vec{a} \vec{b} \vec{d} ] [ \vec{c} \vec{e} \vec{f}] - [ \vec{a} \vec{b} \vec{c} ] [ \vec{d} \vec{e} \vec{f}] \)
- Option (b): \([ \vec{a} \vec{b} \vec{e} ] [ \vec{f} \vec{c} \vec{d}] - [ \vec{a} \vec{b} \vec{f} ] [ \vec{e} \vec{c} \vec{d}] \)
- Option (c): \([ \vec{c} \vec{d} \vec{a} ] [ \vec{b} \vec{e} \vec{f}] - [ \vec{a} \vec{d} \vec{b} ] [ \vec{a} \vec{e} \vec{f}] \)
- Option (d): \([ \vec{a} \vec{c} \vec{e} ] [ \vec{b} \vec{d} \vec{f}] \)

After analyzing the options, we find that option (b) matches the expanded expression we derived.

5. **Conclusion**:
Therefore, the correct answer is:
\[ \text{(b) } [ \vec{a} \vec{b} \vec{e} ] [ \vec{f} \vec{c} \vec{d}] - [ \vec{a} \vec{b} \vec{f} ] [ \vec{e} \vec{c} \vec{d}] \]