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# ｍ：ｎ の内分のベクトル表示

$3\displaystyle{{ \text{OR} }\limits^{\longrightarrow }=\frac{\hspace{2}n\hspace{2}}{\hspace{2} m+n \hspace{2}}{p}\limits^{\rightarrow }+\frac{\hspace{2}m\hspace{2}}{\hspace{2} m+n \hspace{2}}{q}\limits^{\rightarrow }}$

となる．

## ■導出

$3\displaystyle{\begin{array}{lll} { \text{OR} }\limits^{\longrightarrow } & ={ \text{OP} }\limits^{\longrightarrow }+{ \text{PR} }\limits^{\longrightarrow } & \vspace{6}\\ & ={ \text{OP} }\limits^{\longrightarrow }+\frac{\hspace{2}m\hspace{2}}{\hspace{2} m+n \hspace{2}}{ \text{PQ} }\limits^{\longrightarrow } & \vspace{6}\\ & ={ \text{OP} }\limits^{\longrightarrow }+\frac{\hspace{2}m\hspace{2}}{\hspace{2} m+n \hspace{2}} \( { \text{PO} }\limits^{\longrightarrow }+{ \text{OQ} }\limits^{\longrightarrow } \) & \vspace{6}\\ & ={ \text{OP} }\limits^{\longrightarrow }+\frac{\hspace{2}m\hspace{2}}{\hspace{2} m+n \hspace{2}} \( - { \text{OP} }\limits^{\longrightarrow }+{ \text{OQ} }\limits^{\longrightarrow } \) & \vspace{6}\\ & = \( 1- \frac{\hspace{2}m\hspace{2}}{\hspace{2} m+n \hspace{2}} \) { \text{OP} }\limits^{\longrightarrow }+\frac{\hspace{2}m\hspace{2}}{\hspace{2} m+n \hspace{2}}{ \text{OQ} }\limits^{\longrightarrow } & \vspace{6}\\ & =\frac{\hspace{2}n\hspace{2}}{\hspace{2} m+n \hspace{2}}{ \text{OP} }\limits^{\longrightarrow }+\frac{\hspace{2}m\hspace{2}}{\hspace{2} m+n \hspace{2}}{ \text{OQ} }\limits^{\longrightarrow } & \vspace{6}\\ & =\frac{\hspace{2}n\hspace{2}}{\hspace{2} m+n \hspace{2}}{p}\limits^{\rightarrow }+\frac{\hspace{2}m\hspace{2}}{\hspace{2} m+n \hspace{2}}{q}\limits^{\rightarrow } & \vspace{6}\\ \end{array}}$

あるいは，線分OQ上に点$3\displaystyle{{\text{\hspace{5}Q\hspace{5}}}^{\prime }}$$3\displaystyle{\text{OP}\text{//}{\text{\hspace{5}Q\hspace{5}}}^{\prime }\text{\hspace{5}R}}$ に成るように，線分OP上に点$3\displaystyle{{\text{\hspace{5}P}}^{\prime }}$$3\displaystyle{\text{OQ}\text{//}{\text{\hspace{5}P}}^{\prime }\text{\hspace{5}R}}$ に成るようにとると，

$3\displaystyle{\begin{array}{lll} { \text{OR} }\limits^{\longrightarrow } & ={ {\text{OP}}^{\prime } }\limits^{\longrightarrow }+{ {\text{OQ}}^{\prime } }\limits^{\longrightarrow } & \vspace{6}\\ & =\frac{\hspace{2}n\hspace{2}}{\hspace{2} m+n \hspace{2}}{ \text{OP} }\limits^{\longrightarrow }+\frac{\hspace{2}m\hspace{2}}{\hspace{2} m+n \hspace{2}}{ \text{OQ} }\limits^{\longrightarrow } & \vspace{6}\\ & =\frac{\hspace{2}n\hspace{2}}{\hspace{2} m+n \hspace{2}}{p}\limits^{\rightarrow }+\frac{\hspace{2}m\hspace{2}}{\hspace{2} m+n \hspace{2}}{q}\limits^{\rightarrow } & \vspace{6}\\ \end{array}}$

となり，同じ結果になる．（右図参照）

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