Question

If \(\vec{A} = a_1 \hat{i} + a_2 \hat{j} + a_3 \hat{k}\), then \(\hat{i} \times (\hat{i} \times \vec{A})\) is

• A. \(a_2 \hat{j} - a_3 \hat{k}\)
• B. \(a_1 \hat{j} + a_3 \hat{k}\)
• C. \(-a_2 \hat{j} - a_3 \hat{k}\)
• D. \(a_3 \hat{j} - a_2 \hat{k}\)

Hint:

Vector triple products are best solved using the identity \(\vec{a} \times (\vec{b} \times \vec{c})\).

Answer 1

The Correct Option is C

Step 1: Use the vector triple product identity.
\[ \vec{a} \times (\vec{b} \times \vec{c}) = \vec{b}(\vec{a}\cdot\vec{c}) - \vec{c}(\vec{a}\cdot\vec{b}) \]
Step 2: Apply the identity to the given expression.
\[ \hat{i} \times (\hat{i} \times \vec{A}) = \hat{i}(\hat{i}\cdot\vec{A}) - \vec{A}(\hat{i}\cdot\hat{i}) \]
Step 3: Evaluate dot products.
\[ \hat{i}\cdot\vec{A} = a_1, \quad \hat{i}\cdot\hat{i} = 1 \]
Step 4: Substitute and simplify.
\[ = a_1\hat{i} - (a_1\hat{i} + a_2\hat{j} + a_3\hat{k}) \] \[ = -a_2\hat{j} - a_3\hat{k} \]
Step 5: Conclusion.
The correct result is \(-a_2 \hat{j} - a_3 \hat{k}\).