If $\vec{a}, \vec{b}$ and $\vec{c}$ are unit vectors satisfying\ $|\vec{a}-\vec{b}|^{2}+|\vec{b}-\v

Question

If $\vec{a}, \vec{b}$ and $\vec{c}$ are unit vectors satisfying\ $|\vec{a}-\vec{b}|^{2}+|\vec{b}-\vec{c}|^{2}+|\vec{c}-\vec{a}|^{2}=9$, then $|2 \vec{a}+5 \vec{b}+5 \vec{c}|$ is\

Professor PC Thomas Classes · Answer

(3) : We have $|\vec{a}-\vec{b}|^{2}+|\vec{b}-\vec{c}|^{2}+|\vec{c}-\vec{a}|^{2}$\ $=3\left\{|a|^{2}+|b|^{2}+|c|^{2}ight\}-(\vec{a}+\vec{b}+\vec{c})^{2}$\ $\Rightarrow \quad 3 	imes 3=(\vec{a}+\vec{b}+\vec{c})^{2}=9$\ $\Rightarrow \quad \vec{a}+\vec{b}+\vec{c}=0 \quad 	herefore \vec{b}+\vec{c}=-\vec{a}$\ $	herefore \quad|2 \vec{a}+5 \vec{b}+5 \vec{c}|=|2 \vec{a}+5(-\vec{a})|=3|\vec{a}|=3$\ [ $\because \vec{a}$ is a unit vector]\