I have the following points in $\mathbb{P}^3(\mathbb{R})$: $A=[1, 0, 1, 1]$, $B=[1, 1, 0, -1]$, $C=[1, 1, 1, 1]$, $D=[0, 1, 0, 1]$, $E=[-1, 2, 3, 2]$. I am asked to find the intersection of the line $AB$ and of the plane $(CDE)$ (I know that generally a plane means a subspace of dimension $2$, so I think that this means that in $\mathbb{R}^4$ $CDE$ should be a subspace of dimension $3$, which is a hyperplane).
So, the first thing that I must do is to find the equation of these two. I am quite sure I know how to find the equation of $(CDE)$. I will put $(CDE):c_1x_1+c_2x_2+c_3x_3+c_4x_4=0$ and after I plug in the coordinates of $C$, $D$ and $E$ I should be able to find those constants $c_i$ I guess and I should be done.
However, I have no idea how I am supposed to find the equation of $AB$. If I approach it the same way, then I will end up with having to find $8$ constants. Obviously, I don't have enough equations. So, what should I do? I have little to none experience working in both $\mathbb{R}^4$ and projective geometry.
Recall from geometry in 3-space that, given points $P_1, P_2 \in \mathbb{R}^3$, the line between them is given by $$ \{tP_1 + (1-t)P_2 : t \in \mathbb{R}\} \, . $$ Similarly, the line between $A$ and $B$ is given by \begin{align*} \{\lambda [1:0:1:1] &+ \mu [1:1:0:-1] : [\lambda:\mu] \in \mathbb{P}^1\}\\ &= \{[\lambda+\mu:\mu:\lambda:\lambda-\mu] : [\lambda:\mu] \in \mathbb{P}^1\} \, . \end{align*} You can see that this indeed passes through $A$ and $B$ by setting $[\lambda:\mu] = [1:0]$ and $[0:1]$. (In fact, you could use exactly the same parametrization from elementary geometry as above, just by taking $t$ to be the affine coordinate $t = \lambda/\mu$ and then homogenizing. This would give a different, but equivalent parametrization.)
If you want an implicit equation for the line as the intersection of two planes, in general you'll have to do some linear algebra as described in the linked answers, but in this example you can just read off the answer. Setting $$ [X_0 : X_1 : X_2 : X_3] = [\lambda+\mu:\mu:\lambda:\lambda-\mu] \, , $$ we see that these points satisfy $$ X_0 = X_1 + X_2 \qquad X_3 = X_2 - X_1 \, . $$