doubts about finding an equation of a projective plane

Question

Let a,b,c point of projective space $\mathbb P^3(\mathbb R)$ of homogeneus coordinates $a=[1,A]$ $b=[1,B]$ $c=[1,C]$ find cartesian equation of projective plane containing $a$, $b$ and $c$. This is the original test but i have some doubts , isn't $\mathbb P^3(\mathbb R)$ isomorphic to $\mathbb R^4$? I think this problem is impossible but I could be wrong. Thanks to everyone

Answer 1

No, $\mathbb P^3(\mathbb R)$ is not isomorphic to $\mathbb R^4$, but you can imagine $\mathbb P^3(\mathbb R)$ as $\mathbb R^4$, where all points on a line through the origin are glued together, i.e. we identify two points $x\neq 0$ and $y\neq 0$ in $\mathbb R^4$, if there is a $\lambda\in \mathbb R$ with $y=\lambda x$.

What do you think about $$ \det\begin{bmatrix}1 & x & y & z\\1&a_1&a_2&a_3\\ 1&b_1&b_2&b_3\\ 1&c_1&c_2&c_3\end{bmatrix}=0? $$