Calculate the hyperplane of ${\rm I\!R}^4$

Question

Calculate the hyperplane of ${\rm I\!R}^4$ which goes through the points $P1 = (1, 2, 1, 4)$,

$P2 = (0, 1, 2, 1)$, $P3 = (-2, 1, 1, 0)$ y $P4 = (1, 1, 1, 0)$.

I try:

\begin{vmatrix} x & y & z & t & 1 \\ 1 & 2 & 1 & 4 & 1 \\ 0 & 1 & 2 & 1 & 1 \\ -2 & 1 & 1 & 0 & 1 \\ 1 & 1 & 1 & 0 & 1 \end{vmatrix}

but I do not know if it's correct.

Answer 1

The equation of the plane is $$ax+by+cz+dw+e=0$$ now substitute the coordinates of the points $P$ to get a system of linear equations.

Well, this is equivalent to your solution, if you use Cramer to solve the system.