If A, B, C, D are any four points and E and F | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

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

Vedclass · Accepted Answer

$4\overline{EF}$

Vedclass · Answer

(B) Let the position vectors of points $A, B, C, D$ be $\vec{a}, \vec{b}, \vec{c}, \vec{d}$ respectively.Since $E$ is the midpoint of $AC$,its position vector is $\vec{e} = \frac{\vec{a} + \vec{c}}{2}$,so $\vec{a} + \vec{c} = 2\vec{e}$.Since $F$ is the midpoint of $BD$,its position vector is $\vec{f} = \frac{\vec{b} + \vec{d}}{2}$,so $\vec{b} + \vec{d} = 2\vec{f}$.We need to evaluate the sum $\vec{S} = \overline{AB} + \overline{CB} + \overline{CD} + \overline{AD}$.In terms of position vectors,this is:$\vec{S} = (\vec{b} - \vec{a}) + (\vec{b} - \vec{c}) + (\vec{d} - \vec{c}) + (\vec{d} - \vec{a})$$\vec{S} = 2\vec{b} + 2\vec{d} - 2\vec{a} - 2\vec{c}$$\vec{S} = 2(\vec{b} + \vec{d}) - 2(\vec{a} + \vec{c})$Substituting the midpoint relations:$\vec{S} = 2(2\vec{f}) - 2(2\vec{e})$$\vec{S} = 4\vec{f} - 4\vec{e} = 4(\vec{f} - \vec{e}) = 4\overline{EF}$.

If $A, B, C, D$ are any four points and $E$ and $F$ are the midpoints of $AC$ and $BD$ respectively,then $\overline{AB} + \overline{CB} + \overline{CD} + \overline{AD} = \dots$

Similar Questions

Let $ABCDEF$ be a regular hexagon with the vertices $A, B, C, D, E, F$ in counter-clockwise order. If $O$ is the center of $ABCDEF$,then the vector $\vec{AO}$ is equal to which of the following?

$P$ is a point on the side $BC$ of the $\Delta ABC$ and $Q$ is a point such that $\overrightarrow{PQ}$ is the resultant of $\overrightarrow{AP}, \overrightarrow{PB}, \overrightarrow{PC}$. Then $ABQC$ is a

If $|\overline{u}|=2$ and $\overline{u}$ makes angles of $60^{\circ}$ and $120^{\circ}$ with axes $OX$ and $OY$ respectively,then $\overline{u}=$

Suppose $ABCDE$ is a pentagon. The resultant vector of the vectors $\vec{AB}, \vec{AE}, \vec{BC}, \vec{DC}, \vec{ED}$ and $\vec{AC}$ is

Let $u, v$ and $w$ be three vectors in $R^3$. Then,any vector $z \in R^3$ can be written as $z = au + bv + cw$ for some scalars $a, b$ and $c$ if and only if:

Vedclass Test Series

Exam Paper Generator

Online Exam Module