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: Approach
Use the dot product inequality (Cauchy-Schwarz).

Step 2: Inequality
$(\vec u\times\vec v)\cdot\vec w\le|\vec u\times\vec v|\,|\vec w|=|\vec u\times\vec v|$, with equality when $\vec w$ is along the cross product.

Step 3: Compute
Components: $(2\cdot1-(-2)\cdot0,\ (-2)\cdot2-1\cdot1,\ 1\cdot0-2\cdot2)=(2,-5,-4)$. Length $\sqrt{45}=3\sqrt5$.

Step 4: Answer
Option (C).

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