Question

For any vector, \(\bar{a}\) if \(\hat{i}\times (\bar{a}\times \hat{i})+\hat{j}\times (\bar{a}\times \hat{j})+\hat{k}\times (\bar{a}\times \hat{k})=m\bar{a}\), then the value of \(m\) is ...

• A. \(2\)
• B. \(3\)
• C. \(1\)
• D. \(-1\)

Hint:

Expand each term with the vector triple product formula and add them.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept
The triple product formula is \(\bar{p}\times (\bar{a}\times \bar{p})=\bar{a}(\bar{p}\cdot \bar{p})-\bar{p}(\bar{p}\cdot \bar{a})\).

Step 2: Expand
For a unit vector \(\hat{i}\): \(\hat{i}\times (\bar{a}\times \hat{i})=\bar{a}-(\bar{a}\cdot \hat{i})\hat{i}\).
Similarly for \(\hat{j}\) and \(\hat{k}\).

Step 3: Add
The sum is \(3\bar{a}-\left[(\bar{a}\cdot \hat{i})\hat{i}+(\bar{a}\cdot \hat{j})\hat{j}+(\bar{a}\cdot \hat{k})\hat{k}\right]\).

Step 4: Simplify
The bracket equals \(\bar{a}\), so the sum is \(3\bar{a}-\bar{a}=2\bar{a}\). Hence \(m=2\).

Final Answer:
The value of \(m\) is \(2\). \[ \boxed{\text{(A) }2} \]