Translations

In comparison to rotating reference frames, translations are much simpler. Here we consider two reference frames, A and B, that have the same orientation, but their origins, A_o and B_o, are no longer coincident. The position of point, P, relative to B_o is denoted by the vector ^B\bold{r}{P/Bo}. The leading superscript is to denote that this vector is expressed in the _B frame. In other words,

\mathbf{^{B}r_{P/B_{0}} = r_{B_{x}}\hat{b}_{x} + r_{B_{y}}\hat{b}_{y} + r_{B_{z}}\hat{b}_{z}}

The goal then is to describe P relative to A_o, ^A\bold{r}_{P/Ao}

Because both frames have the same relative orientation, describing P relative to A_o only requires simple vector addition:

\mathbf{^{A}r_{P/A_{0}} = ^{A}r_{B_{0}/A_{0}} + ^{B}r_{P/B_{0}} }

where ^A\bold{r}{Bo/Ao} is the origin of the _B frame relative to the origin of the A frame. It is important to note that the location of point P has not changed in any way, we have simply described its position relative to a different frame of reference.