If \( \vec{a} \) and \( \vec{b} \) are two vectors such that \( |\vec{a}|=2,\ |\vec{b}|=1 \) and \( \vec{a}.\vec{b}=\sqrt{3} \) then the angle between \( 2\vec{b} \) and \( -\vec{a} \) is:
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → \( \displaystyle\frac{5\pi}{6} \)
Let \( \theta \) be the angle between \( \vec{a} \) and \( \vec{b} \):
\( \displaystyle \cos\theta=\frac{\vec{a}\cdot\vec{b}}{|\vec{a}|\,|\vec{b}|} \)
\( \displaystyle =\frac{\sqrt{3}}{2} \)
\( \displaystyle \theta=\frac{\pi}{6} \)
Multiplying \( \vec{b} \) by 2 does not change its direction, but \( -\vec{a} \) points opposite to \( \vec{a} \). So the angle is
\( \displaystyle \pi-\frac{\pi}{6}=\frac{5\pi}{6} \)
Why the other options are incorrect
Option (1) \( \displaystyle\frac{\pi}{6} \) — Incorrect. This is the angle between \( \vec{a} \) and \( \vec{b} \); the minus sign reverses the direction of \( \vec{a} \).
Option (2) \( \displaystyle\frac{\pi}{3} \) — Incorrect. The angle is \( \frac{5\pi}{6} \), not \( \frac{\pi}{3} \).
Option (4) \( \displaystyle\frac{5\pi}{3} \) — Incorrect. The angle between two vectors lies between 0 and \( \pi \).