I have a line defined as $k + \langle t\rangle$ and a plane defined as $s + \langle u, v\rangle$ where $k, t, s, u, v$ are 4-dimensional vectors (I know the coordinates). I need to find the distance between this plane and this line.
I also don't really know how to work in 4-dimensional space. In a 3-dimensional space I'd check whether the line and the plane are parallel by finding a vector orthogonal to both $u$ and $v$ and checking whether it is orthogonal to $t$. If they are not, the distance is 0, otherwise I'd try to form a general plane equation (the $Ax + By + Cz + D = 0$ one), take any point of the line and calculate the answer.
But I'm really confused about doing something of that sort in a 4-dimensional space. I don't even know how to check whether the line and the plane are parallel because I can find a vector orthogonal to $u$ and $v$ which is not orthogonal to $t$, but I can also find a vector which is orthogonal to all three of them. Then I don't know how to form a general plane equation: with 3 dimensions I'd form a parametric equation and calculate the determinant of a 3x3 matrix, but I can't do that in 4 dimensions, I'd get a 4x3 matrix.