If `a=1/sqrt(10)(3i+k)" and "b=1/7(2i+3j-6k)`, then the value of `(2a-b).[(a times b) times (a+2b)]` is

Options

A. 3
B. -5
C. -3
D. 5

Answer

To solve the problem, we need to compute the expression \((2\mathbf{a} - \mathbf{b}) \cdot [(\mathbf{a} \times \mathbf{b}) \times (\mathbf{a} + 2\mathbf{b})]\).

### Step-by-Step Solution:

1. **Define the Vectors**:
Given:
\[ \mathbf{a} = \frac{1}{\sqrt{10}}(3\mathbf{i} + \mathbf{k}), \quad \mathbf{b} = \frac{1}{7}(2\mathbf{i} + 3\mathbf{j} - 6\mathbf{k}) \]

2. **Calculate \(2\mathbf{a} - \mathbf{b}\)**:
\[ 2\mathbf{a} = 2 \cdot \frac{1}{\sqrt{10}}(3\mathbf{i} + \mathbf{k}) = \frac{2}{\sqrt{10}}(3\mathbf{i} + \mathbf{k}) = \frac{6}{\sqrt{10}}\mathbf{i} + \frac{2}{\sqrt{10}}\mathbf{k} \]
\[ \mathbf{b} = \frac{1}{7}(2\mathbf{i} + 3\mathbf{j} - 6\mathbf{k}) = \frac{2}{7}\mathbf{i} + \frac{3}{7}\mathbf{j} - \frac{6}{7}\mathbf{k} \]
Now, compute \(2\mathbf{a} - \mathbf{b}\):
\[ 2\mathbf{a} - \mathbf{b} = \left(\frac{6}{\sqrt{10}} - \frac{2}{7}\right)\mathbf{i} - \frac{3}{7}\mathbf{j} + \left(\frac{2}{\sqrt{10}} + \frac{6}{7}\right)\mathbf{k} \]

3. **Calculate \(\mathbf{a} \times \mathbf{b}\)**:
Using the determinant method:
\[ \mathbf{a} \times \mathbf{b} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ \frac{3}{\sqrt{10}} & 0 & \frac{1}{\sqrt{10}} \\ \frac{2}{7} & \frac{3}{7} & -\frac{6}{7} \end{vmatrix} \]
Calculate the determinant to find \(\mathbf{a} \times \mathbf{b}\).

4. **Calculate \(\mathbf{a} + 2\mathbf{b}\)**:
\[ 2\mathbf{b} = \frac{2}{7}(2\mathbf{i} + 3\mathbf{j} - 6\mathbf{k}) = \frac{4}{7}\mathbf{i} + \frac{6}{7}\mathbf{j} - \frac{12}{7}\mathbf{k} \]
\[ \mathbf{a} + 2\mathbf{b} = \frac{1}{\sqrt{10}}(3\mathbf{i} + \mathbf{k}) + \left(\frac{4}{7}\mathbf{i} + \frac{6}{7}\mathbf{j} - \frac{12}{7}\mathbf{k}\right) \]

5. **Calculate \((\mathbf{a} \times \mathbf{b}) \times (\mathbf{a} + 2\mathbf{b})\)**:
Use the vector triple product identity:
\[ \mathbf{x} \times (\mathbf{y} \times \mathbf{z}) = (\mathbf{x} \cdot \mathbf{z})\mathbf{y} - (\mathbf{x} \cdot \mathbf{y})\mathbf{z} \]

6. **Final Dot Product**:
Now compute the dot product:
\[ (2\mathbf{a} - \mathbf{b}) \cdot [(\mathbf{a} \times \mathbf{b}) \times (\mathbf{a} + 2\mathbf{b})] \]