Composition of Two Translations

The composition of two translations is simply another translation.

Suppose that the point $P$ has position vector $\vec{p}$. We can let $P$ be any point we wish.

Let the translation $T_1$ be defined by the vector $\vec{u}$; that is, it shifts every point by that vector $\vec{u}$. Similarly, let the translation $T_2$ be defined by the vector $\vec{v}$.

Let us denote $Q=T_1(P)$ and $R=T_2(Q)$. Thus, we have $R=T_2\circ T_1(P)$.

Since the point $P$ is shifted by the vector $\vec{u}$, the position vector of $Q$ is $\vec{p}+\vec{u}$.

Since the point $Q$ is shifted by the vector $\vec{v}$, the position vector of $R$ is $\vec{p}+\vec{u}+\vec{v}=\vec{p}+(\vec{u}+\vec{v})$.

This means that we have shifted the point $P$ by the vector $\vec{u}+\vec{v}$. Since the point $P$ was an arbitrary choice, the same holds for all points. Thus, the composition $T_2\circ T_1$ is another translation defined by the vector $\vec{u}+\vec{v}$.

Notice that we could have applied the transformation $T_2$ first, and then $T_1$ afterward; since vector addition is commutative, this would have given us the same result. Thus, when you have a composition of translations, one may apply the translations in any order you want.