Find |a-b|, if two vectors a and b are such that |a|=2,|b|=3 and … - Vidyadip

Question

Find $|\vec{a}-\vec{b}|$, if two vectors $\vec a$ and $\vec b$ are such that $|\vec{a}|=2,|\vec{b}|=3$ and $\vec{a} \cdot \vec{b}=4$.

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Answer

We have
$|\vec{a}-\vec{b}|^{2}=(\vec{a}-\vec{b}) \cdot(\vec{a}-\vec{b})$
= $\vec{a} \cdot \vec{a}-\vec{a} \cdot \vec{b}-\vec{b} \cdot \vec{a}+\vec{b} \cdot \vec{b}$
= $|\vec{a}|^{2}-2(\vec{a} \cdot \vec{b})+|\vec{b}|^{2}$
= $(2)^{2}-2(4)+(3)^{2}$
Therefore, $|\vec{a}-\vec{b}|=\sqrt{5}$

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