If C is the midpoint of overline{AB} and P is | vedclass

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(B) Let the position vectors of points $A$,$B$,and $C$ be $\vec{a}$,$\vec{b}$,and $\vec{c}$ respectively. Since $C$ is the midpoint of $\overline{AB}$

Vedclass · Accepted Answer

$\vec{PA} + \vec{PB} = 2\vec{PC}$

Vedclass · Answer

(B) Let the position vectors of points $A$,$B$,and $C$ be $\vec{a}$,$\vec{b}$,and $\vec{c}$ respectively. Since $C$ is the midpoint of $\overline{AB}$,we have $\vec{c} = \frac{\vec{a} + \vec{b}}{2}$,which implies $\vec{a} + \vec{b} = 2\vec{c}$.Let the position vector of point $P$ be $\vec{p}$.Then,$\vec{PA} = \vec{a} - \vec{p}$ and $\vec{PB} = \vec{b} - \vec{p}$.Adding these,we get $\vec{PA} + \vec{PB} = (\vec{a} + \vec{b}) - 2\vec{p}$.Substituting $\vec{a} + \vec{b} = 2\vec{c}$,we get $\vec{PA} + \vec{PB} = 2\vec{c} - 2\vec{p} = 2(\vec{c} - \vec{p})$.Since $\vec{PC} = \vec{c} - \vec{p}$,we have $\vec{PA} + \vec{PB} = 2\vec{PC}$.

If $C$ is the midpoint of $\overline{AB}$ and $P$ is any point not on $\overline{AB}$,then which of the following is true?

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