Rigid body pose - one equation, four unknowns

The master equation, used for every pose question in this section:

p^* = R\,p + r_C

Symbol legend:

$p^*$ - the point in world coordinates
$p$ - the same physical point in body coordinates
$R$ - the body’s orientation in world; its columns are the body’s axes expressed in world coordinates
$r_C$ - the body’s origin (center of mass) in world coordinates

Decision table

Identify what’s being asked, then use that row:

Asked for	Use	Note
Point in world coordinates	$p^* = Rp + r_C$	as-is
Point in body coordinates	$p = R^T(p^* - r_C)$	subtract first, then $R^T$ on the left
Position / origin / center of mass	$r_C = p^* - Rp$	compute $Rp$ first
Orientation $R$	read the columns off the figure	not algebra: column $j$ is the body’s $j$ -th axis in world coordinates
Pose	state both $\{r_C, R\}$	position and orientation together

Label every vector with its frame before starting. Most mistakes are a mislabel, not bad algebra.
$R$ always multiplies a body-frame vector. If $R$ sits next to something, that something is in body coordinates.
Matrix inverses go on the left, because that is the side $R$ sits on. Matrix multiplication does not commute, so you cannot divide across as in scalar algebra.
For rotation matrices $R^{-1} = R^T$ . Always use the transpose; never invert.
Notation: a superscript names the frame the coordinates are expressed in, so $r_p^{W_1}$ is the vector to point $p$ written in frame $W_1$ . Never write a bare vector without its frame.

Rotations preserve length. If a transform changes a distance, it is wrong.
Sketch the frames before computing, then check the algebra agrees with the drawing.