Question

If $\mathbf{a}= -\hat{i} + 2\hat{j} - \hat{k}$, $\mathbf{b} = \hat{i} + \hat{j} - 3\hat{k}$ and $\mathbf{c} = -4\hat{i} - \hat{k}$, then $\mathbf{a} \times (\mathbf{b} \times \mathbf{c}) + (\mathbf{a} \cdot \mathbf{b})\mathbf{c}$ is

• A. $5\hat{i} + 5\hat{j} - 15\hat{k}$
• B. $\mathbf{0}$
• C. $12\hat{j} + 4\hat{k}$
• D. $-3\hat{i} + 6\hat{j} - 3\hat{k}$

Hint:

Vector triple product: $\mathbf{a} \times (\mathbf{b} \times \mathbf{c}) = (\mathbf{a} \cdot \mathbf{c})\mathbf{b} - (\mathbf{a} \cdot \mathbf{b})\mathbf{c}$. Remember: ``BAC minus CAB'' rule.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
Apply the vector triple product identity: $\mathbf{a} \times (\mathbf{b} \times \mathbf{c}) = (\mathbf{a} \cdot \mathbf{c})\mathbf{b} - (\mathbf{a} \cdot \mathbf{b})\mathbf{c}$.
Step 2: Detailed Explanation:
$\mathbf{a} \times (\mathbf{b} \times \mathbf{c}) + (\mathbf{a} \cdot \mathbf{b})\mathbf{c} = (\mathbf{a} \cdot \mathbf{c})\mathbf{b} - (\mathbf{a} \cdot \mathbf{b})\mathbf{c} + (\mathbf{a} \cdot \mathbf{b})\mathbf{c} = (\mathbf{a} \cdot \mathbf{c})\mathbf{b}$.
$\mathbf{a} \cdot \mathbf{c} = (-1)(-4) + (2)(0) + (-1)(-1) = 4 + 0 + 1 = 5$.
Result $= 5\mathbf{b} = 5(\hat{i} + \hat{j} - 3\hat{k}) = 5\hat{i} + 5\hat{j} - 15\hat{k}$.
Step 3: Final Answer:
The result is $5\hat{i} + 5\hat{j} - 15\hat{k}$.