If D, E, F are respectively the midpoints of | vedclass

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

Question

(A) Let the position vectors of vertices $A, B, C$ be $\vec{a}, \vec{b}, \vec{c}$ respectively.Since $D, E, F$ are midpoints of $AB, AC, BC$ respectiv

Vedclass · Accepted Answer

$\overrightarrow{DC}$

Vedclass · Answer

(A) Let the position vectors of vertices $A, B, C$ be $\vec{a}, \vec{b}, \vec{c}$ respectively.Since $D, E, F$ are midpoints of $AB, AC, BC$ respectively,their position vectors are:$\vec{d} = \frac{\vec{a} + \vec{b}}{2}$,$\vec{e} = \frac{\vec{a} + \vec{c}}{2}$,$\vec{f} = \frac{\vec{b} + \vec{c}}{2}$.Now,$\overrightarrow{BE} = \vec{e} - \vec{b} = \frac{\vec{a} + \vec{c}}{2} - \vec{b} = \frac{\vec{a} + \vec{c} - 2\vec{b}}{2}$.And $\overrightarrow{AF} = \vec{f} - \vec{a} = \frac{\vec{b} + \vec{c}}{2} - \vec{a} = \frac{\vec{b} + \vec{c} - 2\vec{a}}{2}$.Adding these two vectors:$\overrightarrow{BE} + \overrightarrow{AF} = \frac{\vec{a} + \vec{c} - 2\vec{b} + \vec{b} + \vec{c} - 2\vec{a}}{2} = \frac{2\vec{c} - \vec{a} - \vec{b}}{2} = \vec{c} - \frac{\vec{a} + \vec{b}}{2}$.Since $\vec{d} = \frac{\vec{a} + \vec{b}}{2}$,we have $\overrightarrow{BE} + \overrightarrow{AF} = \vec{c} - \vec{d} = \overrightarrow{DC}$.

If $D, E, F$ are respectively the midpoints of $AB, AC$ and $BC$ in $\Delta ABC$,then $\overrightarrow{BE} + \overrightarrow{AF} = $

Similar Questions

If $\vec{a}=\hat{i}+\hat{j}+2 \hat{k}$ and $\vec{b}=2 \hat{i}+\hat{j}+2 \hat{k},$ find the unit vector in the direction of
$(i)$ $6 \vec{b}$
(ii) $2 \vec{a}-\vec{b}$

If $\overline{e}_1, \overline{e}_2$ and $\overline{e}_1+\overline{e}_2$ are unit vectors,then the angle between $\overline{e}_1$ and $\overline{e}_2$ is (in $^{\circ}$)

$ABCD$ is a parallelogram such that $L$ is the mid-point of $BC$. Then,$\vec{AL}$ is equal to:

If the vectors $3\hat{i} - 2\hat{j} + 5\hat{k}$ and $-2\hat{i} + p\hat{j} - q\hat{k}$ are parallel,then find the values of $p$ and $q$.

If $\vec{a} = \langle \frac{1}{3}, \frac{1}{3}, \frac{1}{3} \rangle$,then what is its magnitude?

Vedclass Test Series

Exam Paper Generator

Online Exam Module