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  • Need solution for RD Sharma maths class 12 chapter 22 Algebra of Vectors exercise Fill in the blanks question 19

Need solution for RD Sharma maths class 12 chapter 22 Algebra of Vectors exercise Fill in the blanks question 19

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Answer:-0

Hint:-To solve this equation we use mid-point formula

Given:-If D,E,F are mid points of the side BC,CA \; and \; AB respectively of \Delta ABC  then  \overrightarrow{A D}+\overrightarrow{B E}+\overrightarrow{C \vec{F}}=---                                                                                                                                

Solution:-

In  \triangle A B C, D, E, F are mid pts.                                                                                                      

Let \vec{a}, \vec{b}, \vec{c}  be the position vector of  \vec{A}, \vec{B}, \vec{C}
Using mid point formula.
As D  is a mid point of BC

D=\frac{\vec{b}+\vec{c}}{2}

 Similarly E and F are the mid points of AC and AB respectively                   

\begin{aligned} &\therefore E=\frac{\vec{a}+\vec{c}}{2} \\\\ &F=\frac{\vec{a}+\vec{b}}{2} \end{aligned}

Now,

\overrightarrow{A D}+\overrightarrow{B E}+\overrightarrow{C F}=\overrightarrow{O D}-\overrightarrow{O A}+\overrightarrow{O E}-\overrightarrow{O B}+\overrightarrow{O F}-\overrightarrow{O C}

                                 \begin{aligned} &=\overrightarrow{O D}+\vec{O} E+\overrightarrow{O F}-[\overrightarrow{O A}+\overrightarrow{O B}+\overrightarrow{O C}] \\\\ &=\frac{\vec{b}+\vec{c}}{2}+\frac{\vec{a}+\vec{c}}{2}+\frac{\vec{a}+\vec{b}}{2}-(\vec{a}+\vec{b}+\vec{c}) \end{aligned}

                                 \begin{aligned} &=\frac{2(\vec{a}+\vec{b}+\vec{c})}{2}-(\vec{a}+\vec{b}+\vec{c}) \\\\ &=(\vec{a}+\vec{b}+\vec{c})-(\vec{a}+\vec{b}+\vec{c}) \\\\ &=0 \end{aligned}

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