The value of `hati xx(hatixxa)+hatjxx(hatjxxa) +hatk xx(hatkxxa)` is

Options

A. a
B. `axxhatk`
C. `-2a`
D. `-a`

Explanation

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

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

where \(\mathbf{a} = a_1 \hat{i} + a_2 \hat{j} + a_3 \hat{k}\).

### Step 1: Calculate \(\hat{i} \times \mathbf{a}\)

Using the properties of the cross product:

\[ \hat{i} \times \mathbf{a} = \hat{i} \times (a_1 \hat{i} + a_2 \hat{j} + a_3 \hat{k}) \]

Calculating each term:

- \(\hat{i} \times \hat{i} = 0\)
- \(\hat{i} \times \hat{j} = \hat{k}\)
- \(\hat{i} \times \hat{k} = -\hat{j}\)

Thus,

\[ \hat{i} \times \mathbf{a} = 0 + a_2 \hat{k} - a_3 \hat{j} = a_2 \hat{k} - a_3 \hat{j} \]

### Step 2: Calculate \(\hat{i} \times (\hat{i} \times \mathbf{a})\)

Now, we compute:

\[ \hat{i} \times (a_2 \hat{k} - a_3 \hat{j}) = a_2 (\hat{i} \times \hat{k}) - a_3 (\hat{i} \times \hat{j}) \]

Calculating each term:

- \(\hat{i} \times \hat{k} = -\hat{j}\)
- \(\hat{i} \times \hat{j} = \hat{k}\)

Thus,

\[ \hat{i} \times (a_2 \hat{k} - a_3 \hat{j}) = -a_2 \hat{j} - a_3 \hat{k} \]

### Step 3: Calculate \(\hat{j} \times \mathbf{a}\)

Next, we compute:

\[ \hat{j} \times \mathbf{a} = \hat{j} \times (a_1 \hat{i} + a_2 \hat{j} + a_3 \hat{k}) \]

Calculating each term:

- \(\hat{j} \times \hat{i} = -\hat{k}\)
- \(\hat{j} \times \hat{j} = 0\)
- \(\hat{j} \times \hat{k} = \hat{i}\)

Thus,

\[ \hat{j} \times \mathbf{a} = -a_1 \hat{k} + a_3 \hat{i} \]

### Step 4: Calculate \(\hat{j} \times (\hat{j} \times \mathbf{a})\)

Now, we compute:

\[ \hat{j} \times (-a_1 \hat{k} + a_3 \hat{i}) = -a_1 (\hat{j} \times \hat{k}) + a_3 (\hat{j} \times \hat{i}) \]

Calculating each term:

- \(\hat{j} \times \hat{k} = \hat{i}\)
- \(\hat{j} \times \hat{i} = -\hat{k}\)

Thus,

\[ \hat{j} \times (-a_1 \hat{k} + a_3 \hat{i}) = -a_1 \hat{i} - a_3 \hat{k} \]

### Step 5: Calculate \(\hat{k} \times \mathbf{a}\)

Next, we compute:

\[ \hat{k} \times \mathbf{a} = \hat{k} \times (a_1 \hat{i} + a_2 \hat{j} + a_3 \hat{k}) \]

Calculating each term:

- \(\hat{k} \times \hat{i} = \hat{j}\)
- \(\hat{k} \times \hat{j} = -\hat{i}\)
- \(\hat{k} \times \hat{k} = 0\)

Thus,

\[ \hat{k} \times \mathbf{a} = a_1 \hat{j} - a_2 \hat{i} \]

### Step 6: Calculate \(\hat{k} \times (\hat{k} \times \mathbf{a})\)

Now, we compute:

\[ \hat{k} \times (a_1 \hat{j} - a_2 \hat{i}) = a_1 (\hat{k} \times \hat{j}) - a_2 (\hat{k} \times \hat{i}) \]

Calculating each term:

- \(\hat{k} \times \hat{j} = -\hat{i}\)
- \(\hat{k} \times \hat{i} = \hat{j}\)

Thus,

\[ \hat{k} \times (a_1 \hat{j} - a_2 \hat{i}) = -a_1 \hat{i} + a_2 \hat{j} \]

### Step 7: Combine all results

Now we combine all the results:

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

Substituting the values we calculated:

\[ (-a_2 \hat{j} - a_3 \hat{k}) + (-a_1 \hat{i} - a_3 \hat{k}) + (-a_1 \hat{i} + a_2 \hat{j}) \]

Combining like terms:

\[ -2a_1 \hat{i} - 2a_2 \hat{j} - 2a_3 \hat{k} = -2(a_1 \hat{i} + a_2 \hat{j} + a_3 \hat{k}) = -2 \mathbf{a} \]

### Final Answer

Thus, the value of the expression is:

\[ -2 \mathbf{a} \]