# Vectors - Finding the midpoint

• Mar 9th 2011, 10:32 PM
brumby_3
Vectors - Finding the midpoint

The position vectors of points A, B, D are respectively a, b, -a+2b.

I know -a+2b is equivalent to points (-1,2), but not sure what to do from here. I know how to do it when I know the points of A, but there's no numbers, so that's why I'm confused! Help is much appreciated.
• Mar 9th 2011, 10:40 PM
$\overrightarrow{AD}=(-\underline{a}+2\underline{b})-\underline{a}=2(-\underline{a}+\underline{b})$

if mid point of $\overrightarrow{AD}$ is $E$

then $\overrightarrow{OE}=\overrightarrow{OA}+\overright arrow{AE}=\overrightarrow{OA}+\frac{\overrightarro w{AD}}{2}$
• Mar 9th 2011, 10:50 PM
brumby_3
Quote:

$\overrightarrow{AD}=(-\underline{a}+2\underline{b})-\underline{a}=2(-\underline{a}+\underline{b})$

if mid point of $\overrightarrow{AD}$ is $E$

then $\overrightarrow{OE}=\overrightarrow{OA}+\overright arrow{AE}=\overrightarrow{OA}+\frac{\overrightarro w{AD}}{2}$

So is the answer [2(-a+b)/2] = -a+b or am I not reading it right?
• Mar 10th 2011, 12:20 AM
earboth
Quote:

Originally Posted by brumby_3
So is the answer [2(-a+b)/2] = -a+b or am I not reading it right?

Unfortunately that's wrong.

1. As BAdhi wrote you have to calculate:

$\overrightarrow{OE}= \overrightarrow{OA}+\frac12 \cdot \overrightarrow{AD}$

which simplifies to:

$\overrightarrow{OE}=\vec a + \frac12 (2(-\vec a + \vec b))=\vec b$

2. In short: The staionary vector of the midpoint of a distance is the mean of the startpoint and the endpoint vectors of the distance:

$\overrightarrow{OE}=\dfrac{\vec a + (-\vec a + 2\vec b)}2 = \vec b$