M and N are the midpoints of the diagonals AC | vedclass

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(C) Let the position vectors of vertices $A, B, C, D$ be $\vec{a}, \vec{b}, \vec{c}, \vec{d}$ respectively.Since $M$ is the midpoint of diagonal $AC$,

Vedclass · Accepted Answer

$4 \overrightarrow{MN}$

Vedclass · Answer

(C) Let the position vectors of vertices $A, B, C, D$ be $\vec{a}, \vec{b}, \vec{c}, \vec{d}$ respectively.Since $M$ is the midpoint of diagonal $AC$,its position vector is $\vec{m} = \frac{\vec{a} + \vec{c}}{2}$.Since $N$ is the midpoint of diagonal $BD$,its position vector is $\vec{n} = \frac{\vec{b} + \vec{d}}{2}$.Now,consider the expression $\overrightarrow{AB} + \overrightarrow{AD} + \overrightarrow{CB} + \overrightarrow{CD}$:$= (\vec{b} - \vec{a}) + (\vec{d} - \vec{a}) + (\vec{b} - \vec{c}) + (\vec{d} - \vec{c})$$= 2\vec{b} + 2\vec{d} - 2\vec{a} - 2\vec{c}$$= 2[(\vec{b} + \vec{d}) - (\vec{a} + \vec{c})]$$= 4 \left[ \frac{\vec{b} + \vec{d}}{2} - \frac{\vec{a} + \vec{c}}{2} \right]$$= 4(\vec{n} - \vec{m})$$= 4 \overrightarrow{MN}$

$M$ and $N$ are the midpoints of the diagonals $AC$ and $BD$ respectively of quadrilateral $ABCD$,then $\overrightarrow{AB}+\overrightarrow{AD}+\overrightarrow{CB}+\overrightarrow{CD}=$

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