Question

If $\bar{a} = 4\hat{i} + 3\hat{j} + \hat{k}, \bar{b} = \hat{i} - 2\hat{j} + 2\hat{k}$ then $\bar{a} \times (\bar{a} \times (\bar{a} \times (\bar{a} \times \bar{b}))) =$}

• A. $676\bar{a}$
• B. $676\bar{b}$
• C. $625\bar{a}$
• D. $625\bar{b}$

Hint:

Use the identity for a cross product of a vector with a cross product, and notice that a and b are perpendicular.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept
\(\bar a\times(\bar a\times\bar b)=\bar a(\bar a\cdot\bar b)-\bar b|\bar a|^2\).

Step 2: Dot product and magnitude
\(\bar a\cdot\bar b=4-6+2=0\) and \(|\bar a|^2=16+9+1=26\).

Step 3: Apply repeatedly
\(\bar a\times(\bar a\times\bar b)=-26\bar b\)
\(\bar a\times(\bar a\times(\bar a\times\bar b))=-26(\bar a\times\bar b)\)
\(\bar a\times(\bar a\times(\bar a\times(\bar a\times\bar b)))=-26\,\bar a\times(\bar a\times\bar b)=-26(-26\bar b)=676\bar b\)

Step 4: Compare the options
The result is \(676\bar b\), the second option.

Final Answer:
The expression equals 676 times vector b. \[ \boxed{\text{(B) }676\bar{b}} \]