Interchanging a position between two reference frames?

[ato]

$\vec{r}_a$ is a positional vector from reference frame $a$. What is the position of same point from reference frame $b$ ?
If required, assume position of origin of frame $a$ is $\vec{m}$ and unit point (i.e. $\langle 1,1,1\rangle_a$ ) is $\vec{n}$ from reference frame $b$.

I am studying Kleppner and Kolenkow and this is the first thing I asked myself. Unexpectedly its taking a while to figure it out. So help needed.

 

[WannabeNewton]

Let $O$ denote the origin of frame $A$ and let $r_{OP}$ denote the position vector from $O$ to some point $P$. Now let's say we have a frame $B$ with origin $O'$ (which may or may not be changing with time) in frame $A$; denote by $r_{OO'}$ the position vector from the origin of frame $A$ to that of frame $B$. Here is a crude diagram: http://postimg.org/image/ynt4tab6f/full/

Notice, from the diagram, that $r_{OO'} = r_{OP} + r_{PO'}$ where clearly $r_{PO'} = -r_{O'P}$. This gives you the relationship between the position vector from $O'$ to $P$ to the position vector from $O$ to $P$ in terms of how $O'$ is positioned relative to $O$.

 

[ato]

Let $O$ denote the origin of frame $A$ and let $r_{OP}$ denote the position vector from $O$ to some point $P$. Now let's say we have a frame $B$ with origin $O'$ (which may or may not be changing with time) in frame $A$; denote by $r_{OO'}$ the position vector from the origin of frame $A$ to that of frame $B$. Here is a crude diagram: http://postimg.org/image/ynt4tab6f/full/

Notice, from the diagram, that $r_{OO'} = r_{OP} + r_{PO'}$ where clearly $r_{PO'} = -r_{O'P}$. This gives you the relationship between the position vector from $O'$ to $P$ to the position vector from $O$ to $P$ in terms of how $O'$ is positioned relative to $O$.

No, its wrong to use of addition of vector law to add to vectors from different reference frame. For example consider two frames with same origin at O' but with different oriantitions. According to $r_{OO'} = r_{OP} + r_{PO'}$, the $r_{OO'}$ for each frame would coinsides. But they should not if $r_{O'P}$ for each do not coinsides.