Get translational component of particular matrix multiplication Say I have a $4\times4$ rotation matrix $R$ and $4\times4$ translation matrix $T.$ If I multiply the matrices in this order $T * R$, the translation component of $T$ will not get affected. But if multiply them in order $R * T$, it will get affected by rotation values of $R$.
How can I describe the positional vector of the resulting matrix from $R * T$ multiplication, in terms of the rotational components of $R$ and the translation component of matrix $T?$ Can I describe it as the dot product of respective orientation component of matrix $R$ and the translation component of matrix $T?$
 A: Here's a mathematical description of what's going on that I hope you find helpful.
Let $[x]$ denote the "homogeneous coordinates" of $x$ (that is, if $x = (x_1,x_2,x_3)$, then $[x] = (x_1,x_2,x_3,1)$).  Suppose $T$ is defined to be a translation by the vector $b = (b_1,b_2,b_3)$, and that 
$$
R = \pmatrix{Q & 0_{3 \times 1}\\0 & 1}
$$
where $Q$ is a $3 \times 3$ matrix.  Then $T$ and $R$ are defined so that
$$
T*[x] = [x + b], \qquad R*[x] = [Qx]
$$
The consequences of applying $R$ then $T$ are clear:
$$
T*R*[x] = T*[Qx] = [Qx + b]
$$
As expected, we have the rotation $Q$ followed by a translation by $b$. On the other hand, if we apply $T$ then $R$, we have
$$
R*T*[x] = R*[x + b] = [Q(x+b)] = [Qx + Qb].
$$
In other words, the net effect can be described as the same rotation $Q$ (or $R$ if you prefer) followed by a translation by the vector $Qb$.
You might find the following helpful. In terms of $T$ and $R$, you can access the offset $Qb$ using the fact that
$$
[Qb] = R*T*[\mathbf 0] = R*[b]
$$
where $\mathbf 0$ is $(0,0,0)$.
