How can I find the distance of the intersection between the 2-plane and the given line? Consider $\Bbb R^4$ equipped with the Euclidean inner product, and the hyperplane $H$ described by the equation $$3x − 5y + 2z + w = 3.$$ Furthermore consider the $2$-plane $P$ in $H$ given by the equations \begin{align}4x − 4y − z &= 1,\\ x − 3z + y − w &= −2,\end{align} and the line $L$ in $H$ given by the parametric equation \begin{align}x &= t+2,& y &= t+1,\\ z &= −t−2,& w &= 4t+6.\end{align} What is the distance of the intersection point $S$ of $P$ with $L$ from the point $Q = (−1, −1, 3, −12)$ outside $H$.  
This question has confused me as I am not entirely sure what a $2$-plane means and how I can find the distance from the intersection of $S$ with $P$ when I have two 
planes for $P$?
 A: A "$2$-plane" is a linear structure (or affine structure, depending on your terminology) that has $2$ dimensions. In other words, it is a "plane" in the usual sense, but it may exist in a structure outside our usual $3$-dimensional space. In your case, it exists in $4$-dimensional space. Describing that $2$-plane parametrically would take two independent parameters: that is why it is called $2$-dimensional.
We know that it is $2$-dimensional since it is defined as the intersection of two linearly-independent linear equations. Each linearly-independent equation reduces the size of the solution set by one. So in $4$-space, two equations defines a $4-2$ dimensional structure.
You "have two planes for $P$" because your $2$-plane $P$ is defined as the intersection of the two $3$-planes defined by each equation.
To find the intersection point of line $L$ and plane $P$, just substitute the equations you have for $L$ into the equations you have for $P$. You will get just one value of $t$ that satisfies the two equations, namely $t=-5$. Substitute that back into the equations for line $L$ to get the actual intersection point.
Finally, calculate the distance from that point to your other point $(-1,-1,3,-12)$.
Note that we did not use or need the definition of hyperplane $H$: I assume part of the point of the problem is to see if you can screen out the irrelevant. There is another solution that uses $H$ and ignores $P$, but that is risky since you do not know if the equations for $P$ would mean there is no intersection point.
