A translation operation should move a point, but should have no effect on a direction because a direction does not have a position.

How can this be achieved in $3$D with a homogeneous $44$ translation matrix, $T,$ which encodes a translation by $t_x,t_y,t_z,$ and a $3$D direction vector, $d=[d_x,d_y,d_z]$

$?$

The goal is to have $Td=d$ since a translation, $T,$ should not change a direction, $d.$

Set the fourth component of $d$ to $0$ before applying $T.$

Set the fourth component of $d$ to $1$ before applying $T.$

Set the last row of $T$ to $-t_x,-t_y,-t_z,1$ before applying $T.$

Set the last row of $T$ to $0,0,0,-(t_x+t_y+t_z)$ before applying $T.$