Express a point on a segment as a weighted combination of its endpoints.
Notes and key methods
If \(\displaystyle AM:MB=m:n\), the displacement AM is \(\displaystyle \frac{m}{m+ n}\) of AB. Thus the position vector of M is \(\displaystyle a+ \frac{m(b- a)}{m+ n}= \frac{na+ mb}{m+ n}\).
To prove points collinear, show one connecting displacement is a scalar multiple of another. To prove a point lies on a segment, the multiplier must also lie between zero and one. Independent directions allow coefficient comparisons.
Check the conditions
The stated ratio is AM:MB, in that order.
The two direction vectors used for coefficient comparison are independent.
Three worked examples
Example 1 · Vector ratios and geometric proofs
\(\displaystyle OA=\)\(\left[\begin{matrix}4\\2\end{matrix}\right]\), \(\displaystyle OB=\)\(\left[\begin{matrix}1\\6\end{matrix}\right]\). M divides AB with \(\displaystyle AM:MB=1:5\). For the general ratio formula write \(\displaystyle m=1, n=5\).
Write \(\mathbf a=\overrightarrow{OA}\) and \(\mathbf b=\overrightarrow{OB}\).
Find OM and show that M is between A and B.
Derive the weighted-endpoint formula from AM.
Find the coefficient of a in OM.
Find the coefficient of b in OM.
Show the reasoning
(a)
\(\displaystyle AM= \frac{1}{1+ 5}AB\). This multiplier is strictly between zero and one, so M lies inside the segment.
\(\displaystyle OA=\)\(\left[\begin{matrix}5\\2\end{matrix}\right]\), \(\displaystyle OB=\)\(\left[\begin{matrix}1\\7\end{matrix}\right]\). M divides AB with \(\displaystyle AM:MB=2:2\). For the general ratio formula write \(\displaystyle m=2, n=2\).
Write \(\mathbf a=\overrightarrow{OA}\) and \(\mathbf b=\overrightarrow{OB}\).
Find OM and show that M is between A and B.
Derive the weighted-endpoint formula from AM.
Find the coefficient of a in OM.
Find the coefficient of b in OM.
Show the reasoning
(a)
\(\displaystyle AM= \frac{2}{2+ 2}AB\). This multiplier is strictly between zero and one, so M lies inside the segment.
\(\displaystyle OA=\)\(\left[\begin{matrix}6\\2\end{matrix}\right]\), \(\displaystyle OB=\)\(\left[\begin{matrix}1\\8\end{matrix}\right]\). M divides AB with \(\displaystyle AM:MB=3:3\). For the general ratio formula write \(\displaystyle m=3, n=3\).
Write \(\mathbf a=\overrightarrow{OA}\) and \(\mathbf b=\overrightarrow{OB}\).
Find OM and show that M is between A and B.
Derive the weighted-endpoint formula from AM.
Find the coefficient of a in OM.
Find the coefficient of b in OM.
Show the reasoning
(a)
\(\displaystyle AM= \frac{3}{3+ 3}AB\). This multiplier is strictly between zero and one, so M lies inside the segment.