## 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
$(24/5,49/5)$
.