The three points are$$A(1,-2,1)\qquad B(4,-2,-2)\qquad C(4,1,4)$$The airplane I obtain is$$x+2y+z+6=0$$but that obviously does not pass v the three points $A,B,C$.

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Here"s one means to get the requisite plane:

Get two different vectors which room in the plane, such as $B-A=(3,0,-3)$ and also $C-A=(3,3,3)$.Compute the cross product the the two acquired vectors: $(B-A)×(C-A)=(9,-18,9)$. This is the normal vector that the plane, so we have the right to divide the by 9 and get $(1,-2,1)$.The equation of the aircraft is for this reason $x-2y+z+k=0$. To get $k$, substitute any suggest and solve; we acquire $k=-6$.

The last equation of the airplane is $x-2y+z-6=0$.

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You’re trying to find an equation of the kind $ax+by+cz+d=0$. Plugging the works with of the recognized points into this generic equation offers you the complying with system of straight equations: \beginalign a-2b+c+d&=0 \\ 4a-2b-2c+d&=0 \\ 4a+b+4c+d&=0.\endalign fix this mechanism for the unknown coefficients $a$, $b$, $c$ and also $d$. The equipment won’t it is in unique, however if all goes fine (you haven’t made a mistake and the points aren’t colinear) the solution space will be one-dimensional. That’s to be expected due to the fact that you can multiply the equation the a plane by any type of nonzero continuous to get an additional equation for the same plane.

The above system deserve to be created as the procession equation $$\beginbmatrix1&-2&1&1\\4&-2&-2&1\\4&1&4&1\endbmatrix \beginbmatrixa\\b\\c\\d\endbmatrix = 0$$ indigenous which it’s evident that the coefficients of the equation of the airplane are the materials of any type of nonzero element of the null an are of the matrix on the left. The very first three columns are simply the $x$-, $y$- and also $z$-coordinates the the 3 points, because of this one can discover the equation that the plane through 3 noncolinear point out by computer the null an are of $$\beginbmatrixx_1&y_1&z_1&1\\x_2&y_2&z_2&1\\x_3&y_3&z_3&1\endbmatrix.$$ In fact, it’s feasible to do far better and compose down an equation that the aircraft directly. Every other point $(x,y,z)$ ~ above the airplane also generates a linear equation in the coefficients that the aircraft equation. In order to add it come the above system without reducing the dimension of the equipment set, it need to be dependent on the other equations, i.e., it should be a linear mix of the other three. This way that for any allude $(x,y,z)$ top top the plane, the rows of $$A = \beginbmatrixx&y&z&1\\x_1&y_1&z_1&1\\x_2&y_2&z_2&1\\x_3&y_3&z_3&1\endbmatrix$$ should be linearly dependent, but that method that $\det A=0$ is one equation the the plane. Using this idea to the 3 points in your problem produces the equation $9x-18y+9z-54=0$, which becomes $x-2y+z-6=0$ after ~ eliminating the usual factor of $9$. This method is applicable come a wide variety of curves and surfaces.