Question

A, B, C, D, E are five coplanar points such that $\overrightarrow{DA}+\overrightarrow{DB}+\overrightarrow{DC}+\overrightarrow{AE}+\overrightarrow{BE}+\overrightarrow{CE}=\lambda \overrightarrow{DE},\mathrm{Then}\lambda =\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_.$

Answer 1

Given: $\overrightarrow{DA}+\overrightarrow{DB}+\overrightarrow{DC}+\overrightarrow{AE}+\overrightarrow{BE}+\overrightarrow{CE}=\lambda \overrightarrow{DE}$


$$\begin{gathered} \overrightarrow{DA}+\overrightarrow{DB}+\overrightarrow{DC}+\overrightarrow{AE}+\overrightarrow{BE}+\overrightarrow{CE}=\lambda \overrightarrow{DE} \\ \Rightarrow \left(\overrightarrow{DA}+\overrightarrow{AE}\right)+\left(\overrightarrow{DB}+\overrightarrow{BE}\right)+\left(\overrightarrow{DC}+\overrightarrow{CE}\right)=\lambda \overrightarrow{DE} \\ \Rightarrow \left(\overrightarrow{DE}\right)+\left(\overrightarrow{DE}\right)+\left(\overrightarrow{DE}\right)=\lambda \overrightarrow{DE} \\ \Rightarrow 3\left(\overrightarrow{DE}\right)=\lambda \left(\overrightarrow{DE}\right) \\ \Rightarrow 3=\lambda \end{gathered}$$


Hence, $\lambda$ = 3.