<p>The value of \(\hat{i} \cdot (\hat{j} \times \hat{k}) + \hat{j} \cdot (\hat{k} \times \hat{i}) + \hat{k} \cdot (\hat{i} \times \hat{j})\) is</p>
Step-by-Step Solution
Key Concept: Apply the cyclic property of cross product of standard basis vectors and compute scalar triple products.
Solution: Using the properties of standard basis vectors: \(\hat{j} \times \hat{k} = \hat{i}\), so \(\hat{i} \cdot (\hat{j} \times \hat{k}) = \hat{i} \cdot \hat{i} = 1\) \(\hat{k} \times \hat{i} = \hat{j}\), so \(\hat{j} \cdot (\hat{k} \times \hat{i}) = \hat{j} \cdot \hat{j} = 1\) \(\hat{i} \times \hat{j} = \hat{k}\), so \(\hat{k} \cdot (\hat{i} \times \hat{j}) = \hat{k} \cdot \hat{k} = 1\) Therefore: \(\hat{i} \cdot (\hat{j} \times \hat{k}) + \hat{j} \cdot (\hat{k} \times \hat{i}) + \hat{k} \cdot (\hat{i} \times \hat{j}) = 1 + 1 + 1 = 3\) ∴ Answer is (a).
Correct Answer: A