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Provide solution for RD Sharma maths class 12 chapter 22 Algebra of Vectors exercise Fill in the blanks question 14

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Answer:-  3

Hint:-To solve this equation we do \overrightarrow{A B}=\overrightarrow{O B}-\overrightarrow{O A} method

Given:-A, B, C, D, E are five coplanar, such that \overrightarrow{D A}+\overrightarrow{D B}+\overrightarrow{D C}+\overrightarrow{A E}+\overrightarrow{B E}+\overrightarrow{C E}=K \overrightarrow{D E} \text { then } \mathrm{k} \text { is.... }

Solution:- \overrightarrow{A B}=\overrightarrow{O B}-\overrightarrow{O A}

\overrightarrow{D A}+\overrightarrow{D B}+\overrightarrow{D C}+\overrightarrow{A E}+\overrightarrow{B E}+\overrightarrow{C E}=K \overrightarrow{D E}

\begin{aligned} &\Rightarrow(\overrightarrow{O A}-\overrightarrow{O D})+(\overrightarrow{O B}-\overrightarrow{O D})+(\overrightarrow{O C}-\overrightarrow{O D})+(\overrightarrow{O E}-\overrightarrow{O A})+(\overrightarrow{O E}-\overrightarrow{O B})+(\overrightarrow{O E}-\overrightarrow{O C})=k \overrightarrow{D E} \\\\ &\Rightarrow 3(\overrightarrow{O E}-\overrightarrow{O D})=k \overrightarrow{D E} \\ \end{aligned}

\Rightarrow 3 \overrightarrow{D E}=k \overrightarrow{D E} \\

\Rightarrow 3=k \\

\Rightarrow k=3

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