# determine the point of intersection on a facet in n-dimensions

I'm trying to solve a what I think is a classic line/plane intersection problem. However, this type of problem is new to me so please excuse me if I am misusing the terminology. I have two points in 4 dimensional space....

[14000, 14000,24000, 4000]
[475, 10015, 436, 20008]


so the direction or vector between these two points is...

[13525, 3985, 23564, -16008]


I also have a facet or plane that is defined by these four points:

[[1935, 10007, 2200, 18464],
[1013, 10986, 1671, 17678],
[1276, 9460, 1245, 17099],
[3021, 8722, 1890, 19507]]


How do I calculate the point in this facet (or plane) where the vector between the two points would intersect it, if at all? I've seen examples of plane intersection solved using the cross product of two vectors that comprise the plane, but these examples are usually in 3 dimensions and cross product seems to only be used for 3 and 7 dimensions. Is cross product even what I want to use here?

It seems the following.

The equation of the segment between two points $x_1$ and $x_2$ is $$x=\lambda_1 x_1+\lambda_2 x_2,$$ $\lambda_1,\lambda_2\ge 0$, $\lambda_1+\lambda_2=1$. Similarly, the (hyper)plane spanned on four points $y_1$, $y_2$, $y_3$, $y_4$ consists of the point $y$ of the form $$y=\mu_1 y_1+\mu_2 y_2+\mu_3 y_3+\mu_4 y_4,$$ $\mu_1+\mu_2+\mu_3+\mu_4=1$. So to find the intersection between the segment and the plane, we have to solve the following system of six linear equations with six unknowns:

$\mu_1 y_1+\mu_2 y_2+\mu_3 y_3+\mu_4 y_4=\lambda_1 x_1+\lambda_2 x_2$ (applied to coordinates, this equality yield four linear equations),

$\lambda_1+\lambda_2=1$,

$\mu_1+\mu_2+\mu_3+\mu_4=1$,

with the restriction

$\lambda_1,\lambda_2\ge 0$.

• I'm new to this type of mathematics, so I don't understand what some of the greek symbols mean. I'm guessing that lambda is a scalar, right? Is mu also a scalar for the each point on the hyperplane? – b10hazard Jan 26 '15 at 14:00
• @b10hazard Yes. – Alex Ravsky Jan 26 '15 at 20:13