The vector
`hati xx [(axxb) xx hati] + hatj xx [(axxb)xxhatj ] + hatk xx [(axxb) xxhatk]` is equal

Options

A. `2( axxb)`
B. b
C. `(a.b)b`
D. 0

Answer

To solve the vector expression

\[ \hat{i} \times [(a \times b) \times \hat{i}] + \hat{j} \times [(a \times b) \times \hat{j}] + \hat{k} \times [(a \times b) \times \hat{k}] \]

we will use the vector triple product identity, which states:

\[ \mathbf{u} \times (\mathbf{v} \times \mathbf{w}) = (\mathbf{u} \cdot \mathbf{w}) \mathbf{v} - (\mathbf{u} \cdot \mathbf{v}) \mathbf{w} \]

### Step 1: Break down each term using the vector triple product identity.

For the first term, we have:

\[ \hat{i} \times [(a \times b) \times \hat{i}] \]

Using the identity:

\[ \hat{i} \times [(a \times b) \times \hat{i}] = (\hat{i} \cdot \hat{i}) (a \times b) - (\hat{i} \cdot (a \times b)) \hat{i} \]

Since \(\hat{i} \cdot \hat{i} = 1\), this simplifies to:

\[ = (a \times b) - (\hat{i} \cdot (a \times b)) \hat{i} \]

### Step 2: Calculate the second term.

For the second term:

\[ \hat{j} \times [(a \times b) \times \hat{j}] \]

Using the identity again:

\[ \hat{j} \times [(a \times b) \times \hat{j}] = (\hat{j} \cdot \hat{j}) (a \times b) - (\hat{j} \cdot (a \times b)) \hat{j} \]

Since \(\hat{j} \cdot \hat{j} = 1\), this simplifies to:

\[ = (a \times b) - (\hat{j} \cdot (a \times b)) \hat{j} \]

### Step 3: Calculate the third term.

For the third term:

\[ \hat{k} \times [(a \times b) \times \hat{k}] \]

Using the identity:

\[ \hat{k} \times [(a \times b) \times \hat{k}] = (\hat{k} \cdot \hat{k}) (a \times b) - (\hat{k} \cdot (a \times b)) \hat{k} \]

Since \(\hat{k} \cdot \hat{k} = 1\), this simplifies to:

\[ = (a \times b) - (\hat{k} \cdot (a \times b)) \hat{k} \]

### Step 4: Combine all terms.

Now, we combine all three terms:

\[ \hat{i} \times [(a \times b) \times \hat{i}] + \hat{j} \times [(a \times b) \times \hat{j}] + \hat{k} \times [(a \times b) \times \hat{k}] \]

This gives:

\[ [(a \times b) - (\hat{i} \cdot (a \times b)) \hat{i}] + [(a \times b) - (\hat{j} \cdot (a \times b)) \hat{j}] + [(a \times b) - (\hat{k} \cdot (a \times b)) \hat{k}] \]

### Step 5: Simplify the expression.

Combining the like terms, we have:

\[ 3(a \times b) - [(\hat{i} \cdot (a \times b)) \hat{i} + (\hat{j} \cdot (a \times b)) \hat{j} + (\hat{k} \cdot (a \times b)) \hat{k}] \]

Let \(c = a \times b\), then:

\[ = 3c - (c \cdot \hat{i}) \hat{i} - (c \cdot \hat{j}) \hat{j} - (c \cdot \hat{k}) \hat{k} \]

### Step 6: Final Result.

The expression simplifies to:

\[ = 2(a \times b) \]

Thus, the final answer is:

\[ \boxed{2(a \times b)} \]