I am working on a vision application and therefore I need to find the $x$ and $y$ values when $z=0$ for an object that is moving linearly (constant speed).
So what would $x$ and $y$ for point 3 be, if we know that $z=0$ and the speed between $2\to 3$ is equal to the speed between $1\to2$.
Set up the movement equation: $$\begin{pmatrix}x(t)\\y(t)\\z(t)\end{pmatrix}=\vec{x}(t)=\vec{x_0}+t\cdot \vec{d}=\begin{pmatrix}x_0\\y_0\\z_0\end{pmatrix}+t\cdot \begin{pmatrix}d_x\\d_y\\d_z\end{pmatrix},\tag{1}$$ where $\vec{d}$ is the direction of the object and $\vec{x_0}$ any point it travels through.
Now solve $z(t)=0$ for $t$, i.e. solve $$z(t)=z_0+t\cdot d_z=0$$ and plug in back on the right hand side of (1) to get the values of $x$ and $y$.