# Position vector

• Nov 11th 2008, 04:57 AM
geton
Position vector
OABCDE is a regular hexagon. The points A and B have position vectors a and b respectively, referred to the origin O.
Find, in terms of a and b, the position vectors of C, D and E.

I've no idea how to do this. If I sketch a hexagon where will be C, D, E? And what are their position vectors?
• Nov 11th 2008, 06:59 AM
red_dog
$\displaystyle \overrightarrow{OC}=2\overrightarrow{AB}=2\left(\o verrightarrow{OB}-\overrightarrow{OA}\right)=2\left(\overrightarrow{ b}-\overrightarrow{a}\right)$

$\displaystyle \overrightarrow{OD}=\overrightarrow{OC}+\overright arrow{CD}=2\left(\overrightarrow{b}-\overrightarrow{a}\right)-\overrightarrow{OA}=2\overrightarrow{b}-3\overrightarrow{a}$

$\displaystyle \overrightarrow{OE}=\overrightarrow{OD}+\overright arrow{DE}=\overrightarrow{OD}-\overrightarrow{AB}=$
$\displaystyle =\overrightarrow{OD}-\left(\overrightarrow{OB}-\overrightarrow{OA}\right)=-2\overrightarrow{a}+\overrightarrow{b}$
• Nov 11th 2008, 07:12 AM
Soroban
Hello, geton!

Quote:

$\displaystyle OABCDE$ is a regular hexagon.
The points $\displaystyle A$ and $\displaystyle B$ have position vectors $\displaystyle \vec{a}$ and $\displaystyle \vec{b}$ resp., relative to the origin O.
Find, in terms of $\displaystyle \vec{a}$ and $\displaystyle \vec{b}$, the position vectors of $\displaystyle C, D\text{ and }E.$

Code:

          A  b-a  B           *- - - - -*           /      *  \       a /    *      \         /  * b        \       / *              \     O * - - - - * - - - - * C       \ *      c        /         \  *          /       e \  d *      /           \      *  /           *- - - - -*           E        D

We have: .$\displaystyle \vec{a} = \overrightarrow{OA},\;\;\vec{b} = \overrightarrow{OB}$

Let: .$\displaystyle \vec{c} = \overrightarrow{OC},\;\;\vec{d} = \overrightarrow{OD},\;\;\vec{e} = \overrightarrow{OE}$

Note that: . $\displaystyle \overrightarrow{AB} \:=\:\vec{b}-\vec{a}$

Since $\displaystyle OABCDE$ is a regular hexgon, $\displaystyle \overrightarrow{OC} \:=\: 2\!\cdot\overrightarrow{AB}$

. . Therefore: .$\displaystyle \boxed{\vec{c} \;=\;2(\vec{b} - \vec{a})}$

$\displaystyle \vec{d} \:=\:\overrightarrow{OD} \:=\:\vec{c} + \overrightarrow{CD}$

Since $\displaystyle \vec{c} \:=\:2(\vec{b}-\vec{a})\text{ and }\overrightarrow{CD} \:=\:-\vec{a}$

. . $\displaystyle \vec{d} \:=\:2(\vec{b} - \vec{a}) - \vec{a} \quad\Rightarrow\quad\boxed{ \vec{d} \:=\:2\vec{b} - 3\vec{a}}$

$\displaystyle \vec{e} \:=\:\overrightarrow{OE} \:=\:\vec{d} + \overrightarrow{DE}$

Since $\displaystyle \overrightarrow{DE} \:=\: -\overrightarrow{AB} \:=\:\vec{a}-\vec{b}$

. . $\displaystyle \vec{e} \;=\;(2\vec{b}-3\vec{a}) + (\vec{a} - \vec{b}) \quad\Rightarrow\quad\boxed{ \vec{e} \:=\:\vec{b} - 2\vec{a}}$

• Nov 11th 2008, 04:16 PM
geton
Quote:

Originally Posted by Soroban
Hello, geton!

Code:

          A  b-a  B           *- - - - -*           /      *  \       a /    *      \         /  * b        \       / *              \     O * - - - - * - - - - * C       \ *      c        /         \  *          /       e \  d *      /           \      *  /           *- - - - -*           E        D

We have: .$\displaystyle \vec{a} = \overrightarrow{OA},\;\;\vec{b} = \overrightarrow{OB}$

Let: .$\displaystyle \vec{c} = \overrightarrow{OC},\;\;\vec{d} = \overrightarrow{OD},\;\;\vec{e} = \overrightarrow{OE}$

Note that: . $\displaystyle \overrightarrow{AB} \:=\:\vec{b}-\vec{a}$

Since $\displaystyle OABCDE$ is a regular hexgon, $\displaystyle \overrightarrow{OC} \:=\: 2\!\cdot\overrightarrow{AB}$

. . Therefore: .$\displaystyle \boxed{\vec{c} \;=\;2(\vec{b} - \vec{a})}$

$\displaystyle \vec{d} \:=\:\overrightarrow{OD} \:=\:\vec{c} + \overrightarrow{CD}$

Since $\displaystyle \vec{c} \:=\:2(\vec{b}-\vec{a})\text{ and }\overrightarrow{CD} \:=\:-\vec{a}$

. . $\displaystyle \vec{d} \:=\:2(\vec{b} - \vec{a}) - \vec{a} \quad\Rightarrow\quad\boxed{ \vec{d} \:=\:2\vec{b} - 3\vec{a}}$

$\displaystyle \vec{e} \:=\:\overrightarrow{OE} \:=\:\vec{d} + \overrightarrow{DE}$

Since $\displaystyle \overrightarrow{DE} \:=\: -\overrightarrow{AB} \:=\:\vec{a}-\vec{b}$

. . $\displaystyle \vec{e} \;=\;(2\vec{b}-3\vec{a}) + (\vec{a} - \vec{b}) \quad\Rightarrow\quad\boxed{ \vec{e} \:=\:\vec{b} - 2\vec{a}}$

Thank you so much :)