Question

If \(\overset{⃗}{u} = \hat{i}+2\hat{j}-2\hat{k},\overset{⃗}{v} = 2\hat{i}+\hat{k}\) and \(\overset{⃗}{w}\) is unit vector then the maximum value of scalar triple product \([\overset{⃗}{u} \overset{⃗}{v} \overset{⃗}{w}]\) is

• A. \(-3\sqrt{5}\)
• B. \(0\)
• C. \(3\sqrt{5}\)
• D. \(\sqrt{54}\)

Hint:

The scalar triple product is largest when w is parallel to the cross product of u and v.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept
\([\vec u\ \vec v\ \vec w]=(\vec u\times\vec v)\cdot\vec w\). For a unit vector \(\vec w\), this is largest when \(\vec w\) points along \(\vec u\times\vec v\), and the maximum equals \(|\vec u\times\vec v|\).

Step 2: Cross product
\(\vec u=(1,2,-2)\), \(\vec v=(2,0,1)\).
\[ \vec u\times\vec v=\begin{vmatrix}\hat i&\hat j&\hat k\\1&2&-2\\2&0&1\end{vmatrix}=(2,-5,-4) \]

Step 3: Magnitude
\[ |\vec u\times\vec v|=\sqrt{4+25+16}=\sqrt{45}=3\sqrt5 \]

Step 4: Conclusion
The maximum value is \(3\sqrt5\), option (C). The value \(-3\sqrt5\) is the minimum, and \(\sqrt{54}\) does not match the cross product magnitude.

Final Answer:
The maximum equals the length of u cross v, which is 3 root 5, option (C). \[ \boxed{3\sqrt{5}} \]