`[vecaxx vecb " " vecc xx vecd " " vecexx vecf]` is equal to

Explanation

To solve the problem `[veca x vecb " " vecc x vecd " " vece x vecf]`, we will follow the steps outlined in the video transcript.

### Step-by-step Solution:

1. **Define the Cross Products**:
- Let \( A = \vec{a} \times \vec{b} \)
- Let \( B = \vec{c} \times \vec{d} \)
- Let \( C = \vec{e} \times \vec{f} \)

2. **Use the Associative Property of Cross Products**:
- The cross product is associative, which means we can rearrange the terms. Thus, we can express the entire expression as:
\[ A \times (B \times C) \]

3. **Apply the Vector Triple Product Identity**:
- We can use the vector triple product identity:
\[ \vec{x} \times (\vec{y} \times \vec{z}) = (\vec{x} \cdot \vec{z}) \vec{y} - (\vec{x} \cdot \vec{y}) \vec{z} \]
Here, let \( \vec{x} = \vec{a} \), \( \vec{y} = \vec{c} \), and \( \vec{z} = \vec{e} \).

4. **Substitute the Values**:
- Substitute \( B = \vec{c} \times \vec{d} \) and \( C = \vec{e} \times \vec{f} \) into the identity:
\[ \vec{a} \times (\vec{c} \times \vec{d}) = (\vec{a} \cdot \vec{d}) \vec{c} - (\vec{a} \cdot \vec{c}) \vec{d} \]

5. **Combine with the Third Cross Product**:
- Now, we need to multiply the result by \( \vec{e} \times \vec{f} \):
\[ \text{Final Result} = \left[(\vec{a} \cdot \vec{d}) \vec{c} - (\vec{a} \cdot \vec{c}) \vec{d}\right] \times (\vec{e} \times \vec{f}) \]

6. **Expand Using the Vector Triple Product Again**:
- We can apply the vector triple product identity again to expand this expression.

7. **Final Expression**:
- After performing the necessary calculations, we can summarize the final expression as:
\[ \text{Final Result} = (\vec{a} \cdot \vec{d})(\vec{c} \times \vec{e} \times \vec{f}) - (\vec{a} \cdot \vec{c})(\vec{d} \times \vec{e} \times \vec{f}) \]

### Final Answer:
The expression `[veca x vecb " " vecc x vecd " " vece x vecf]` simplifies to:
\[ (\vec{a} \cdot \vec{d})(\vec{c} \times \vec{e} \times \vec{f}) - (\vec{a} \cdot \vec{c})(\vec{d} \times \vec{e} \times \vec{f}) \]