<s>[INST] Find an equation of the plane that passes through the points $(1, 2, 5), (5, 4, 8) \textrm{ and } (2, 4, 8)$ Find an equation of the plane that passes through the points $(1, 2, 5), (5, 4, 8) $, and $(2, 4, 8)$.
$v_1 = [1,2,5] - [5,4, 8] = [-4, -2, -3]$
$v_2 = [2,4,8] - [5,4,8] = [-3,0,0]$
$v_1 \times v_2 = [0,9,-6]$
thus,
$ax + by + cz = d, a = 0, b = 9, c = -6$
to get d we plug $0(5) + 9(4) - 6(8) = -12$
Therefore the equation is $9y - 6z = -12$ 
Right?
 [/INST] Another useful formula is
$$
\begin{vmatrix}
 1&1&1&1\\
 1&2&5&x\\
 5&4&8&y\\
 2&4&8&z
\end{vmatrix}=0
$$
We can see that in the 1st row there are 1-s, then the coordinates of the points followed by $x,y,z$, respectively. This is, in fact equivalent to the formula given by the previous answer.
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