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Explain solution RD Sharma class 12 chapter 22 Algebra of Vectors exercise Fill in the blanks question 20 maths

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Answer:- 0

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

Given:-The algebraic sum of the vector directed from the vectors to the mid points of the opposite side is equal to

Solution:-

\begin{aligned} &\text { } \overrightarrow{A D}+\overrightarrow{B \bar{E}}+\overrightarrow{C \vec{F}}\\\\ &\text { In } \Delta A B D\; \; \overrightarrow{A B}+\overrightarrow{B D}+\overrightarrow{D A}=0 \end{aligned}

\begin{aligned} &\overrightarrow{E C}=\frac{1}{2} \overrightarrow{A C} \\\\ &\overrightarrow{B D}=\frac{1}{2} \overrightarrow{B C} \\\\ &\overrightarrow{F B}=\frac{1}{2} \overrightarrow{A B} \end{aligned}

\begin{aligned} &\therefore \overrightarrow{A D}=\overrightarrow{A B}+\frac{1}{2} \overrightarrow{B C} \\\\ &\text { In } \Delta B E C \; \; \overrightarrow{B C}+\overrightarrow{C E}+\overrightarrow{E B}=0 \end{aligned}

\begin{aligned} &\therefore \overrightarrow{B E}=\overrightarrow{B C}+\frac{1}{2} \overrightarrow{C A} \\\\ &\text { In } \Delta A C F \overrightarrow{A F}+\overrightarrow{F C}+\overrightarrow{C A}=0 \\\\ &\therefore \overrightarrow{C F}=\overrightarrow{C A}+\frac{1}{2} \overrightarrow{A B} \end{aligned}

\begin{aligned} &\overrightarrow{A B}+\frac{1}{2} \overrightarrow{B C}+\overrightarrow{B C}+\frac{1}{2} \overrightarrow{C A}+\overrightarrow{C A}+\frac{1}{2} \overrightarrow{A B} \\\\ &\Delta A B C \\\\ &\overrightarrow{A B}+\overrightarrow{B C}+\overrightarrow{C A}=0 \end{aligned}

\overrightarrow{A B}+\overrightarrow{B C}+\overrightarrow{C A}+\frac{1}{2}(\overrightarrow{B C}+\overrightarrow{C A}+\overrightarrow{A B})=0

 

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