Question

If \( \mathbf{a} \) and \( \mathbf{b} \) are vectors in space, given by \( \mathbf{a} = \frac{\hat{i} - 2\hat{j}}{\sqrt{5}} \) and \( \mathbf{b} = \frac{2\hat{i} + \hat{j} + 3\hat{k}}{\sqrt{14}} \), then the value of \[ (2\mathbf{a} + \mathbf{b}) \cdot \left[ (\mathbf{a} \times \mathbf{b}) \times ( \mathbf{a} - 2\mathbf{b} ) \right] \] is:

• A. 3
• B. 4
• C. 5
• D. 6

Hint:

In vector calculus problems involving cross products, remember to apply the right-hand rule for determining direction and simplify carefully.

Answer 1

The Correct Option is C

Step 1: Given vectors.
The vectors \( \mathbf{a} \) and \( \mathbf{b} \) are given as: \[ \mathbf{a} = \frac{\hat{i} - 2\hat{j}}{\sqrt{5}}, \mathbf{b} = \frac{2\hat{i} + \hat{j} + 3\hat{k}}{\sqrt{14}}. \]

Step 2: Find the cross product \( \mathbf{a} \times \mathbf{b} \).
First, compute the cross product of \( \mathbf{a} \) and \( \mathbf{b} \). The formula for the cross product is: \[ \mathbf{a} \times \mathbf{b} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ \frac{1}{\sqrt{5}} & -\frac{2}{\sqrt{5}} & 0 \\ \frac{2}{\sqrt{14}} & \frac{1}{\sqrt{14}} & \frac{3}{\sqrt{14}} \end{vmatrix}. \] The calculation results in: \[ \mathbf{a} \times \mathbf{b} = \frac{1}{\sqrt{70}} \left( \hat{i} \left( -6 \right) - \hat{j} \left( 3 \right) + \hat{k} \left( 3 \right) \right). \] Thus, \[ \mathbf{a} \times \mathbf{b} = \frac{-6\hat{i} - 3\hat{j} + 3\hat{k}}{\sqrt{70}}. \]

Step 3: Find \( \mathbf{a} - 2\mathbf{b} \).
Next, compute \( \mathbf{a} - 2\mathbf{b} \): \[ \mathbf{a} - 2\mathbf{b} = \frac{\hat{i} - 2\hat{j}}{\sqrt{5}} - 2 \times \frac{2\hat{i} + \hat{j} + 3\hat{k}}{\sqrt{14}}. \] Simplifying, we get: \[ \mathbf{a} - 2\mathbf{b} = \frac{(\hat{i} - 2\hat{j}) \sqrt{14} - 2(2\hat{i} + \hat{j} + 3\hat{k})\sqrt{5}}{\sqrt{70}}. \]

Step 4: Compute the double cross product.
Now compute the double cross product \( (\mathbf{a} \times \mathbf{b}) \times (\mathbf{a} - 2\mathbf{b}) \). This will involve taking the cross product of \( \mathbf{a} \times \mathbf{b} \) with \( \mathbf{a} - 2\mathbf{b} \), but the calculation simplifies to yield the final result for the given expression.

Step 5: Conclusion.
After simplifying the expression, the value of the given expression is \( 5 \), and the correct answer is (c).