<s>[INST] How to add constraints to vectors Is it possible to add constraints to vectors, more specifically, planes? For instance:

In this example, is it possible to constrain the red plane with Cartesian equation $z=0$ to only occupy within the area within the area of the yellow lines, such that only the green line intersects the plane and not the purple one?
 [/INST] So you want to parametrize the plane such that a point on the plane is defined by two parameters $$ \vec{r} = \mathbf{ S}(u,v) $$
and then set limits to $u$ and $v$ to define your bounds. In case of a plane, you need a local coordinate system with two mutually orthogonal directions $\mathbf{e}_1$ and $\mathbf{e}_2$ that are also orthogonal to the plane normal $ \mathbf{n} \cdot \mathbf{e}_i = 0$. The origin center is at $\vec{r}_0$.
$$ \vec{r} = \vec{r}_0+ \mathbf{e}_1 u + \mathbf{e}_2 v $$
Now the question becomes: how to set a local coordinate system on a plane defined by the equation $ax +by + cz+d =0 $
The plane normal is $$\mathbf{n} = \frac{1}{\sqrt{a^2+b^2+c^2}} \pmatrix{a\\b\\c}$$ and the point on the plane closest to the origin is
$$ \vec{r}_0 = \pmatrix{ \frac{-a d}{a^2+b^2+c^2} \\ \frac{-b d}{a^2+b^2+c^2} \\ \frac{-c d}{a^2+b^2+c^2} } $$
The choice of local unit vector directions is somewhat arbitrary, but it makes sense to use something related to the bounds you want to define. The only requirement for $\mathbf{e}_1 = \pmatrix{e_x & e_y & e_z}$ is that it has $e_x a + e_y b + e_z c = 0$
One way to achieve this is by finding the direction closest to the x-axis
$$\mathbf{e}_1 = \pmatrix{ \frac{b^2+c^2}{a^2+b^2+c^2} \\ \frac{-a b}{a^2+b^2+c^2} \\ \frac{-a c}{a^2+b^2+c^2} }$$
or the y-axis
$$\mathbf{e}_1 = \pmatrix{ \frac{-a b}{a^2+b^2+c^2} \\ \frac{a^2+c^2}{a^2+b^2+c^2} \\ \frac{-b c}{a^2+b^2+c^2} }$$
or the z-axis
$$\mathbf{e}_1 = \pmatrix{ \frac{-a c}{a^2+b^2+c^2} \\ \frac{-b c}{a^2+b^2+c^2} \\ \frac{a^2+b^2}{a^2+b^2+c^2} }$$
and finally, define the second direction from the cross product of $\mathbf{n}$ and the first direction
$$ \mathbf{e}_2 = \text{unitvector}( \mathbf{n} \times \mathbf{e}_1 ) $$
Summary
With the above scheme you can go from $u$ and $v$ to a point on the plane $\vec{r}=\mathbf{S}(u,v)$. Your limits are set in terms of $(u,v)$ as planar coordinates. For example $u=1$ will produce a line on the plane which is offset from the plane origin by one unit along the local v direction.
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