\[A(2,3)\]
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 \[(24/5,49/5)\]
.