If D, E and F are respectively mid-points of | vedclass

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

Question

(A) Given that $D, E$,and $F$ are the mid-points of $AB, AC$,and $BC$ respectively in $	riangle ABC$.Let the position vectors of vertices $A, B$,and

Vedclass · Accepted Answer

$\overrightarrow{DC}$

Vedclass · Answer

(A) Given that $D, E$,and $F$ are the mid-points of $AB, AC$,and $BC$ respectively in $	riangle ABC$.Let the position vectors of vertices $A, B$,and $C$ be $\vec{a}, \vec{b}$,and $\vec{c}$ respectively.Then,the position vectors of the mid-points are:$\overrightarrow{D} = \frac{\vec{a} + \vec{b}}{2}$$\overrightarrow{E} = \frac{\vec{a} + \vec{c}}{2}$$\overrightarrow{F} = \frac{\vec{b} + \vec{c}}{2}$Now,we calculate the vectors $\overrightarrow{BE}$ and $\overrightarrow{AF}$:$\overrightarrow{BE} = \overrightarrow{E} - \overrightarrow{B} = \frac{\vec{a} + \vec{c}}{2} - \vec{b} = \frac{\vec{a} + \vec{c} - 2\vec{b}}{2}$$\overrightarrow{AF} = \overrightarrow{F} - \overrightarrow{A} = \frac{\vec{b} + \vec{c}}{2} - \vec{a} = \frac{\vec{b} + \vec{c} - 2\vec{a}}{2}$Adding these two vectors:$\overrightarrow{BE} + \overrightarrow{AF} = \frac{\vec{a} + \vec{c} - 2\vec{b} + \vec{b} + \vec{c} - 2\vec{a}}{2}$$= \frac{2\vec{c} - \vec{a} - \vec{b}}{2} = \vec{c} - \frac{\vec{a} + \vec{b}}{2}$Since $\overrightarrow{DC} = \overrightarrow{C} - \overrightarrow{D} = \vec{c} - \frac{\vec{a} + \vec{b}}{2}$,we have:$\overrightarrow{BE} + \overrightarrow{AF} = \overrightarrow{DC}$

If $D, E$ and $F$ are respectively mid-points of $AB, AC$ and $BC$ in $\triangle ABC$,then $\overrightarrow{BE} + \overrightarrow{AF}$ is equal to

Similar Questions

Find a vector in the direction of vector $\vec{a} = \hat{i} - 2\hat{j}$ that has magnitude $7$ units.

$ABC$ is an isosceles triangle right-angled at $A$. Forces of magnitude $2\sqrt{2}$,$5$,and $6$ act along $\overrightarrow{BC}$,$\overrightarrow{CA}$,and $\overrightarrow{AB}$ respectively. The magnitude of their resultant force is

If the collinear points $A, B$ and $C$ have position vectors respectively $(1, x, 3), (3, 4, 7)$ and $(y, -2, -5)$, then $x+y=$

If a unit vector lies in the $yz$-plane and makes angles of $30^\circ$ and $60^\circ$ with the positive $y$-axis and $z$-axis respectively,then its components along the coordinate axes are:

Classify the following measure as a scalar or a vector: $20 \, m/s^{2}$.

Vedclass Test Series

Exam Paper Generator

Online Exam Module