Translation matrix without homogeneous coordinates I am looking to translate a vector, and I found this article on wikipedia, which states that I need to use homogeneous coordinates to do so.
I don't get the reason why, and I don't really see how the result of the translation matrix with homogeneous coordinates is different from a translation matrix
$$ \matrix{T} =
\begin{bmatrix}
v_x & 0 & 0 \\ 
0 & v_y & 0 \\
0 & 0 & v_z \\
\end{bmatrix}
$$
so that with my vector $\vec{p} = [p_x \, p_y \, p_z]^T$ I'd get the translation as
$$
\vec{r}=\matrix{T}\vec{p}=
\begin{bmatrix}
v_x+p_x & 0 & 0 \\ 
0 & v_y+p_y & 0 \\
0 & 0 & v_z+p_z \\
\end{bmatrix}
$$
What is the reason for using the homogeneous coordinates instead of the above?

You see that, in your notation, you would get
$$
T\vec{p} =  \begin{bmatrix}\nu_x p_x \\ \nu_y p_y \\ \nu_z p_z\end{bmatrix}
$$
which is a dilation not a translation.
More generally linear mappings keep the zero vector fixed, while a translation moves it. So you need a way to represent affine transformations.
