Given:
- \(\mathbf{a} = \frac{1}{\sqrt{10}} (4\hat{i} - 3\hat{j} + \hat{k})\)
- \(\mathbf{b} = \frac{1}{5} (\hat{i} + 2\hat{j} + 2\hat{k})\)
Step 1: Calculate \( 2\mathbf{a} - \mathbf{b} \)
First, calculate \( 2\mathbf{a} \):
\( 2\mathbf{a} = 2 \times \frac{1}{\sqrt{10}} (4\hat{i} - 3\hat{j} + \hat{k}) = \frac{2}{\sqrt{10}} (4\hat{i} - 3\hat{j} + \hat{k}) \)
Subtract \( \mathbf{b} \) from \( 2\mathbf{a} \):
\( 2\mathbf{a} - \mathbf{b} = \frac{2}{\sqrt{10}} (4\hat{i} - 3\hat{j} + \hat{k}) - \frac{1}{5} (\hat{i} + 2\hat{j} + 2\hat{k}) \)
Step 2: Calculate \( \mathbf{a} \times \mathbf{b} \)
Next, calculate the cross product \( \mathbf{a} \times \mathbf{b} \) using the determinant:
\( \mathbf{a} \times \mathbf{b} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ \frac{4}{\sqrt{10}} & \frac{-3}{\sqrt{10}} & \frac{1}{\sqrt{10}} \\ \frac{1}{5} & \frac{2}{5} & \frac{2}{5} \end{vmatrix} \)
Simplify the determinant to find \( \mathbf{a} \times \mathbf{b} \).
Step 3: Calculate the triple cross product
Now, compute the triple cross product \( (\mathbf{a} \times \mathbf{b}) \times (\mathbf{a} + 2\mathbf{b}) \). Simplify the vector cross products and perform the dot product between \( (2\mathbf{a} - \mathbf{b}) \) and the result of the triple cross product.
Final Answer:
The simplified value of the expression is \( \boxed{-3} \).
Answer: The correct answer is option (2): -3.