If `a=hat(i)+2hat(j)-2hat(k), b=2hat(i)-hat(j)+hat(k) and c=hat(i)+3hat(j)-hat(k)`, then `atimes(btimesc)` is equal to

Options

A. `20hat(i)-3hat(j)+7hat(k)`
B. `20hat(i)-3hat(j)-7hat(k)`
C. `20hat(i)+3hat(j)-7hat(k)`
D. None of these

Answer

To solve the problem, we need to compute the vector triple product \( \mathbf{a} \times (\mathbf{b} \times \mathbf{c}) \). The formula for the vector triple product is given by:

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

### Step 1: Define the vectors
Given:
\[ \mathbf{a} = \hat{i} + 2\hat{j} - 2\hat{k} \]
\[ \mathbf{b} = 2\hat{i} - \hat{j} + \hat{k} \]
\[ \mathbf{c} = \hat{i} + 3\hat{j} - \hat{k} \]

### Step 2: Calculate \( \mathbf{a} \cdot \mathbf{b} \)
Using the dot product formula:
\[ \mathbf{a} \cdot \mathbf{b} = (1)(2) + (2)(-1) + (-2)(1) \]
Calculating this:
\[ = 2 - 2 - 2 = -2 \]

### Step 3: Calculate \( \mathbf{a} \cdot \mathbf{c} \)
Now, calculate \( \mathbf{a} \cdot \mathbf{c} \):
\[ \mathbf{a} \cdot \mathbf{c} = (1)(1) + (2)(3) + (-2)(-1) \]
Calculating this:
\[ = 1 + 6 + 2 = 9 \]

### Step 4: Substitute into the vector triple product formula
Now substitute the dot products into the formula:
\[ \mathbf{a} \times (\mathbf{b} \times \mathbf{c}) = (9) \mathbf{b} - (-2) \mathbf{c} \]
This simplifies to:
\[ = 9\mathbf{b} + 2\mathbf{c} \]

### Step 5: Calculate \( 9\mathbf{b} \) and \( 2\mathbf{c} \)
Calculating \( 9\mathbf{b} \):
\[ 9\mathbf{b} = 9(2\hat{i} - \hat{j} + \hat{k}) = 18\hat{i} - 9\hat{j} + 9\hat{k} \]

Calculating \( 2\mathbf{c} \):
\[ 2\mathbf{c} = 2(\hat{i} + 3\hat{j} - \hat{k}) = 2\hat{i} + 6\hat{j} - 2\hat{k} \]

### Step 6: Add the two results
Now, combine the results:
\[ \mathbf{a} \times (\mathbf{b} \times \mathbf{c}) = (18\hat{i} - 9\hat{j} + 9\hat{k}) + (2\hat{i} + 6\hat{j} - 2\hat{k}) \]
Calculating this:
\[ = (18 + 2)\hat{i} + (-9 + 6)\hat{j} + (9 - 2)\hat{k} \]
\[ = 20\hat{i} - 3\hat{j} + 7\hat{k} \]

### Final Answer
Thus, the result of \( \mathbf{a} \times (\mathbf{b} \times \mathbf{c}) \) is:
\[ \boxed{20\hat{i} - 3\hat{j} + 7\hat{k}} \]