Corollary 1.1.3.
If P be the mid - point of line AB, then \(m = n\) and \(\vec{OP}
=\frac{1}{2} (\vec{a}+\vec{b})\text{.}\)
Subsection 1.1.3 Dividing the line segments
Let the point P divides the line AB in the ratio of \(m:n\text{.}\) If \(\vec{a}\) \(\vec{b}\) and be the position vectors of points A and B, respectively then the position vector of point P is determined as \(\vec{OA} = \vec{a}\) and \(\vec{OB}= \vec{b}\text{.}\) Since point P divides the line AB in the ratio of \(m:n\text{,}\) as shown in Figure 1.1.2.(c), we have
\begin{align*}
\dfrac{\vec{AP}}{\vec{PB}} \amp = \dfrac{m}{n}\\
\text{or,} \quad n \vec{AP} \amp = m \vec{PB}\\
\text{or,} \quad n\big (\vec{OP}-\vec{OA}\big) \amp = m \big (\vec{OB}-\vec{OP}\big)\\
\text{or,} \quad (m + n)\vec{OP}\amp = m\vec{OB}+n\vec{OA}\\
\text{or,} \quad \vec{OP} \amp = \frac{n\vec{a}+m\vec{b}}{m+n}
\end{align*}
