I am confused about the following proof from a book.
In this figure, $P$ divides $AB$ internally in the ratio $m:n,$ i.e $AP : PB = m : n$ ----(1)
$OP = OA + AP$ ---- (1) and $OP = OB + BP$ ---- (2)
$nOP = nOA + nAP$ ---- (3) and $mOP = mOB + mBP$ ---- (4)
Adding (3) and (4), we get:
$(m+n)OP=mOB+nOA+mBP+nAP$ ---- (5)
This result holds for all values of $m$ and $n$.
Here, $P$ divides $AB$ internally and we take both $m$ and $n$ to be positive.
How do we decide to take $m$ and $n$ to be positive?
Since AP and BP are in opposite directions and $AP : PB = m : n$
$nAP = mPB$
Therefore, $(m+n)OP=mOB+nOA$ ---- (6)
$$OP=\frac{mOB+nOA}{m+n}.$$
Consider fig 3.1(a) and fig 3.1(b)
$AP = (m/n)PB$ and $AP$ and $BP$ are in the same direction.
So, if the ratio $m:n$ is taken to be negative,
How and why are we taking the ratio $m:n$ to be negative?
the expression $mBP+nAP$ in equation (6) will be zero. This as before we get:
$$OP=\frac{mOB+nOA}{m+n}$$