Transformation from one basis to another

Question

I have a 3D point $X_1$ in basis $1$. The origin of basis $1$, $O_1$, has a translation of 't' with respect the origin of basis $2$, $O_2$. Also basis $1$ has a relative rotation with basis $2$, given by the rotation matrix $R$.
How do I go about converting $X_1$ from basis $1$ to basis $2$ ?

Answer 1

Rotate, translate

Call the target vector $Q$. Rotate by the angle $\theta$ in the first coordinate system. $$ R Q. $$

Translate the rotated vector to the second reference frame: $$ \tilde{Q} = RQ + P. $$

A pictorial follows. The black vectors mark the first coordinate system. The blue vectors mark a coordinate system which has been rotated by $\theta$, and translated by a vector $P$.

The target vector $Q$ is located in the first coordinate system and the steps to transform to the second coordinate system follow.

First, rotate the vector by the angle $\theta$.

Translate the rotated vector by $P$.