Translating a point with homogeneous coordinates \((x, y, z, w)\) by a vector \((\Delta x, \Delta y, \Delta z, 0)\) can be computed using the following matrix multiplication:
\[ \begin{bmatrix}
1 & 0 & 0 & \Delta x \\
0 & 1 & 0 & \Delta y \\
0 & 0 & 1 & \Delta z \\
0 & 0 & 0 & 1 \\
\end{bmatrix}
\cdot
\begin{bmatrix}
x \\
y \\
z \\
w \\
\end{bmatrix}\]
Note that translation only has any effect on points (\(w = 1\)): vectors (\(w = 0\)) remain unaffected.