Question

If \(\bar{a},\bar{b}\) are unit vectors and \(\bar{a}\cdot \bar{b}=0\), then \((\bar{a}-2\bar{b})\cdot \{(2\bar{a}+\bar{b})\times (\bar{a}\times \bar{b})\}=...\)

• A. \(5\)
• B. \(6\)
• C. \(8\)
• D. \(10\)

Hint:

Expand the vector triple product using u x (v x w) = v(u.w) - w(u.v) and use a.b = 0 for unit vectors.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept
Given \(|\bar a|=|\bar b|=1\) and \(\bar a\cdot\bar b=0\).

Step 2: Expand the triple product
Use \(\bar u\times(\bar v\times\bar w)=\bar v(\bar u\cdot\bar w)-\bar w(\bar u\cdot\bar v)\) with \(\bar u=2\bar a+\bar b\), \(\bar v=\bar a\), \(\bar w=\bar b\):
\((2\bar a+\bar b)\times(\bar a\times\bar b)=\bar a\,[(2\bar a+\bar b)\cdot\bar b]-\bar b\,[(2\bar a+\bar b)\cdot\bar a]\).

Step 3: Evaluate the dots
\((2\bar a+\bar b)\cdot\bar b=1\) and \((2\bar a+\bar b)\cdot\bar a=2\).
So the cross product equals \(\bar a-2\bar b\).

Step 4: Final dot product
\((\bar a-2\bar b)\cdot(\bar a-2\bar b)=|\bar a|^2+4|\bar b|^2-4\bar a\cdot\bar b=1+4-0=5\).

Final Answer:
The value of the expression is 5. \[ \boxed{\text{(A) }5} \]