Question

If \(\overset{̄}{a} = 2\hat{i}+\hat{j}-\hat{k}\), \(\overset{̄}{b} = \hat{i}+3\hat{k}\) and \(\overset{̄}{c}\) is a unit vector, then the maximum value of the scalar triple product \([\overset{̄}{a} \overset{̄}{b} \overset{̄}{c}]\) is

• A. \(\sqrt{10}+\sqrt{6}\)
• B. \(\sqrt{10}\)
• C. \(\sqrt{6}\)
• D. \(\sqrt{59}\)

Hint:

The triple product equals (a x b) dot c, which is largest when c is along a x b.

Answer 1

The Correct Option is D

Step 1: Geometric meaning
The triple product is the volume of a parallelepiped with base area $|\vec a\times\vec b|$ and height equal to the component of $\vec c$ normal to the base.

Step 2: Maximum height
For a unit vector, the largest height is $1$, reached when $\vec c$ is perpendicular to both $\vec a$ and $\vec b$. So the maximum volume equals the base area $\sqrt{59}$. Option (D).

Final Answer:
root 59. \[ \boxed{\text{(D)}\ \sqrt{59}} \]