You have a point $x$, vector $v$, and also some vertical vector, pointing down. Lets call it $g$. If I have understood your problem correctly, your plane contains $(0,0,0)$ (or otherwise you can write $x$ in the coordinate system with origin on the plane), vector $v$, and a vector $w=v\times g$ --- horizontal vector, orthogonal to $v$. I want to interpret $x$ as a vector (from the origin to the point $x$). Then for some $\alpha\in\mathbb{R}$ vector $x-\alpha g$ is on the plane and $x$ is below the plane if and only if $\alpha>0$. We have
$$(x-\alpha g, v\times w)=0,$$
$$(x-\alpha g, v\times[v\times g])=0,$$
$$(x-\alpha g, v(v,g)-g(v,v))=0,$$
$$(x, v)(v,g)-(x,g)(v,v)=\alpha\bigl((g,v)(g,v)-(g,g)(v,v)\bigr),$$
$$\alpha=\bigl((x, v)(v,g)-(x,g)(v,v)\bigr)/\bigl((g,v)(g,v)-(g,g)(v,v)\bigr).$$