Proof related to vectors: collinear points, position vectors

Question

I'm wondering if the following sentence is correct:

"Given 3 collinear points (A,B,C) and the relative distances (m and n), the position vector OC of the central point is mOA+nOB."

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EDIT: mAB+nAB=AB --> m+n=1.

Thank you so much for your time.

Answer 1

You're right, except that you have $m$ and $n$ backwards. When $m=0$, we want to get $C=A$, so the correct expression is $C=nA+mB$. This equation expresses $C$ as an affine combination of $A$ and $B$.

Answer 2

You have $m$ and $n$ reversed. The correct expression is:

\begin{align} OC &= OA + mAB \\ &=OA+m(OB-OA) \\ &=(1-m)OA + mOB\\ &=nOA+mOB \end{align}

If you can’t remember where the $m$ and $n$ go in this equation, it’s easy to check the cases $m=0$ and $m=1$.