How can I find a hyperplane that passes through a point with arbitrary dimensions? I have a point that contains a known number of arbitrary dimensions. How can I find a hyperplane that passes through that point and the origin?
For what it's worth, for my particular application all the coordinates of the point will be non-negative. If you could phrase your answer in terms of an algorithm, that would be helpful.

Here's an easy way to do it:
Let $P = (a_1,\dots,a_n)$ be the point of interest.  The coordinates of $P$ give us a vector $\vec v = P-(0,\dots,0)$ pointing away from the origin to $P$.  Our plane should contain the origin, and the vector $\vec v$ should be parallel to the plane.  So, the vector $\vec v$ should be perpendicular to the normal vector.
If either $a_1$ or $a_2$ is non-zero, we can make our normal vector
$$
\vec n = \langle -a_2,a_1, 0,\dots,0\rangle
$$
In general, we can make a normal vector by switching any two entries (not both zero), changing the sign of one of the two that we've switched, and setting all other coordinates to $0$.
The equation for the plane with normal vector $n$ through the origin is given by
$$
\vec n \cdot \vec x = 0
$$
Or, for the vector I've given,
$$
-a_2 x_1 + a_1 x_2 = 0
$$
and this is the equation of a hyperplane satisfying the criteria.
