Dividing a Line Segment in a Given Ratio

Suppose the line between  
\[A(2,3)\]
  and  
\[B((9,20)\]
  is to be divided in the ratio  
\[2:3\]
.
Ket the point that divides the line segment in this ratio be  
\[P\]
.
The vector from  
\[A\]
  to  
\[B\]
  is  
\[\vec{AB}=\vec{OB}=\vec{OA}= \begin{pmatrix}9\\20\end{pmatrix}-\begin{pmatrix}2\end{pmatrix}=\begin{pmatrix}7\\17\end{pmatrix}\]
.
The vector  
\[\vec{AP}=\frac{2}{5} \vec{AB}=\frac{2}{5}\begin{pmatrix}7\\17\end{pmatrix}=\begin{pmatrix}14/5\\34/5\end{pmatrix}\]
.
Then  
\[\vec{OP}=\vec{OA}+ \vec{AP}=\begin{pmatrix}2\\3\end{pmatrix}+\begin{pmatrix}14/5\\34/5\end{pmatrix}=\begin{pmatrix}24/5\\49/5\end{pmatrix}\]
.
The point  
\[P\]
  is  
\[(24/5,49/5)\]
.

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