`(vecaxxvecb).(veccxxvecd)` is not equal to

Options

A. `veca.{vecbxx(veccxxvecd)}`
B. `{(vecaxxvecb)xxvecc}vecd`
C. `(vecd xx vecc).(vecbxxveca)`
D. `(veca.vecc)(vecb.vecd)-(veca.vecd)(vecb.vecc)`

Explanation

To solve the problem `(vec a x vec b) . (vec c x vec d)` and determine which of the provided options it is not equal to, we can follow these steps:

### Step 1: Understand the Expression
We start with the expression `(vec a x vec b) . (vec c x vec d)`. This is a dot product of two cross products.

### Step 2: Apply the Vector Triple Product Identity
Using the vector triple product identity, we can express the dot product of two cross products as:
\[ (vec a x vec b) . (vec c x vec d) = (vec c . vec a)(vec b . vec d) - (vec c . vec b)(vec a . vec d) \]
This identity helps us simplify the expression.

### Step 3: Rearranging the Expression
We can rearrange the expression as follows:
\[ (vec a x vec b) . (vec c x vec d) = (vec c . vec a)(vec b . vec d) - (vec c . vec b)(vec a . vec d) \]

### Step 4: Analyze the Options
Now, we need to analyze the options given in the question to find which one does not equal the expression we derived. The options would typically involve combinations of the vectors involved.

### Step 5: Identify the Incorrect Option
From the analysis, we find that the expression is equal to a combination of dot products of the vectors involved. We can check each option against our derived expression to see which one does not fit.

### Conclusion
After evaluating the options, we conclude that option B is the correct answer, as it does not equal the expression `(vec a x vec b) . (vec c x vec d)`.