If a and b are two vectors such that |a|=1,|b|=2 and a b=3, then … - Vidyadip

Question

If $\vec{a}$ and $\vec{b}$ are two vectors such that $|\vec{a}|=1,|\vec{b}|=2$ and $\vec{a} \cdot \vec{b}=\sqrt{3}$, then the angle between $2 \vec{a}$ and $-\vec{b}$ is:

✓

Answer

We have, $|\vec{a}|=1,|\vec{b}|=2$ and $\vec{a} \cdot \vec{b}=\sqrt{3}$
As, $\vec{a} \cdot \vec{b}=|a||b| \cos \theta$
$(2 \vec{a}) \cdot(-\vec{b})=|2 \vec{a}||-\vec{b}| \cos \theta$
$\Rightarrow-2(\vec{a} \cdot \vec{b})=2|\vec{a}||\vec{b}| \cos \theta \Rightarrow \cos \theta=\frac{-(\vec{a} \cdot \vec{b})}{|\vec{a}||\vec{b}|}=\frac{-\sqrt{3}}{2}$
$\therefore \quad$ Angle between $2 \vec{a}$ and $-\vec{b}=\pi-\frac{\pi}{6}$ or $\pi+\frac{\pi}{6}=\frac{5 \pi}{6}$ or $\frac{7 \pi}{6}$

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