1. ## vector example explination.

hello, can some one please explain the example i have attached in detail for each of the steps.
thanks much.

2. Hello, sigma1!

$\displaystyle \begin{Bmatrix}\vec a &=& \overrightarrow{OA} \\ \vec b &=& \overrightarrow{OB} \\ \vec c &=& \overrightarrow{OC}\end{Bmatrix} \qquad |AM| \:=\:|MB|. \qquad D \text{ is on }CM\text{ such that: }\,|CD| \:=\:2\!\cdot\!|DM|$

Code:
              M
A o - - - o - - - o B
|         *   *
|           *
|         *   o D
|       *       *
|     *           *
|   *               *
| *                   *
O o - - - - - - - - - - - o C

$\displaystyle (i)\;\text{ Find the position vector }\overrightarrow{OM} \text{ in terms of }\vec a\text{ and }\vec b.$

. . $\displaystyle \overrightarrow{OM} \;=\;\overrightarrow{OA} + \overrightarrow{AM}$

. . $\displaystyle \boxed{\overrightarrow{OM} \;=\;\vec a + \tfrac{1}{2}\vec b\:}$

$\displaystyle (ii)\;\text{ Find the position vector }\overrightarrow{OD} \text{ in terms of } \vec a, \vec b, \vec c.$

$\displaystyle \overrightarrow{OC} + \overrightarrow{CM} \;=\;\overrightarrow{OA} + \overrightarrow{AM}$

. .$\displaystyle \vec c + \overrightarrow{CM} \;=\;\vec a + \tfrac{1}{2}\vec b$

. . . .$\displaystyle \overrightarrow{CM} \;=\;\vec a + \tfrac{1}{2}\vec b - \vec c$

$\displaystyle \overrightarrow{CD} \;=\;\tfrac{2}{3}\overrightarrow{CM} \;=\;\tfrac{2}{3}\left(\vec a + \tfrac{1}{2}\vec b - \vec c\right) \quad\Rightarrow\quad \overrightarrow{CD} \;=\;\tfrac{2}{3}\vec a + \tfrac{1}{2}\vec b - \tfrac{2}{3}\vec c$ .[1]

$\displaystyle \overrightarrow{OD} \;=\;\overrightarrow{OC} + \overrightarrow{CD} \;=\;\vec c + \underbrace{\left(\tfrac{2}{3}\vec a + \tfrac{1}{3}\vec b - \tfrac{2}{3}\vec c\right)}_{{\color{blue}[1]}}$

. . $\displaystyle \boxed{\overrightarrow{OD} \;=\;\tfrac{2}{3}\vec a + \tfrac{1}{3}\vec b + \tfrac{1}{3}\vec c\:}$

3. i) AM = OM - OA
MB = OB - OM.

Since AM = MB,
OM - OA = OB - OM.
So.
2OM = OA + OB
Or
OM = (a+b)/2.
ii) MD = OD - OM.....(1)
DC = OC - OD = 2MD
MD = OC/2 - OD/2..(2)
Equate (1) and (2)
OD - OM = OC/2 - OD/2
OD+OD/2 = OM + OC/2
3OD/2 = (a+b)/2 + c/2
OD = (a+b+c)/3