Question

If \(\overset{̄}{a},\overset{̄}{b},\overset{̄}{c}\) are three non zero and non-coplanar vectors such that \(\overset{̄}{a}\times (\overset{̄}{b}\times \overset{̄}{c}) = \frac{\overset{̄}{b}}{2}\), then the angle between \(\overset{̄}{a}\) and \(\overset{̄}{b}\) is ...

• A. \(\frac{π}{6}\)
• B. \(\frac{π}{3}\)
• C. \(\frac{π}{2}\)
• D. \(π\)

Hint:

Expand \(\vec a\times(\vec b\times\vec c) = (\vec a\cdot\vec c)\vec b - (\vec a\cdot\vec b)\vec c\) and use non-coplanarity.

Answer 1

The Correct Option is C

Step 1: Key Formula:
\(\vec a\times(\vec b\times\vec c) = (\vec a\cdot\vec c)\,\vec b - (\vec a\cdot\vec b)\,\vec c\).

Step 2: Apply:
Given equality: \((\vec a\cdot\vec c)\vec b - (\vec a\cdot\vec b)\vec c = \frac12\vec b\).
Rearrange: \(\left(\vec a\cdot\vec c - \frac12\right)\vec b - (\vec a\cdot\vec b)\vec c = \vec 0\).

Step 3: Use non-coplanarity:
\(\vec b\) and \(\vec c\) are not parallel (they are part of a non-coplanar set), so the coefficients of both must be zero.
So \(\vec a\cdot\vec b = 0\) and \(\vec a\cdot\vec c = \frac12\).
\(\vec a\cdot\vec b = 0\) with non-zero vectors means the angle between \(\vec a\) and \(\vec b\) is \(\frac{\pi}{2}\).

Final Answer:
The angle between \(\vec a\) and \(\vec b\) is \(\frac{\pi}{2}\), option (C). \[ \boxed{\frac{\pi}{2}} \]