For any vectors `vecr` the value of
`hatixx(vecrxxhati)+hatjxx(vecrxxhatj)+hatkxx(vecrxxhatk)`, is

Options

A. `vec0`
B. `2vecr`
C. `-2vecr`
D. none of these

Explanation

To solve the problem, we need to evaluate the expression:

\[ \hat{i} \times (\vec{r} \times \hat{i}) + \hat{j} \times (\vec{r} \times \hat{j}) + \hat{k} \times (\vec{r} \times \hat{k}) \]

where \(\vec{r}\) is any vector.

### Step 1: Use the Vector Triple Product Identity
We will use the vector triple product identity, which states that:

\[ \vec{a} \times (\vec{b} \times \vec{c}) = \vec{b} (\vec{a} \cdot \vec{c}) - \vec{c} (\vec{a} \cdot \vec{b}) \]

In our case, we can apply this identity to each term in the expression.

### Step 2: Evaluate Each Term
1. For the first term \(\hat{i} \times (\vec{r} \times \hat{i})\):
\[ \hat{i} \times (\vec{r} \times \hat{i}) = \hat{i} (\hat{i} \cdot \hat{i}) - \hat{i} (\vec{r} \cdot \hat{i}) = \hat{i} \cdot 1 - \hat{i} (x) = \hat{i} - x \hat{i} = (1 - x) \hat{i} \]

2. For the second term \(\hat{j} \times (\vec{r} \times \hat{j})\):
\[ \hat{j} \times (\vec{r} \times \hat{j}) = \hat{j} (\hat{j} \cdot \hat{j}) - \hat{j} (\vec{r} \cdot \hat{j}) = \hat{j} \cdot 1 - \hat{j} (y) = \hat{j} - y \hat{j} = (1 - y) \hat{j} \]

3. For the third term \(\hat{k} \times (\vec{r} \times \hat{k})\):
\[ \hat{k} \times (\vec{r} \times \hat{k}) = \hat{k} (\hat{k} \cdot \hat{k}) - \hat{k} (\vec{r} \cdot \hat{k}) = \hat{k} \cdot 1 - \hat{k} (z) = \hat{k} - z \hat{k} = (1 - z) \hat{k} \]

### Step 3: Combine All Terms
Now, we can combine all the terms we calculated:

\[ (1 - x) \hat{i} + (1 - y) \hat{j} + (1 - z) \hat{k} \]

### Step 4: Rewrite in Terms of \(\vec{r}\)
Since \(\vec{r} = x \hat{i} + y \hat{j} + z \hat{k}\), we can express the result as:

\[ \vec{r} - (x \hat{i} + y \hat{j} + z \hat{k}) + \hat{i} + \hat{j} + \hat{k} = \hat{i} + \hat{j} + \hat{k} - \vec{r} \]

### Final Result
Thus, the final result simplifies to:

\[ 2\vec{r} \]

### Conclusion
The value of the expression is:

\[ 2\vec{r} \]