Answer:
x -2y +z = 0
Explanation:
You want to know the equation of the plane through (0, 0, 0), (3, 2, 1), and (1, 1, 1).
Direction
The direction vector can be found as the cross product of (3, 2, 1) and (1, 1, 1). A calculator gives that as (1, -2, 1). This tells you the equation of the plane is ...
1x -2y +1z = c . . . . for some constant c
That constant can be found using any of the point coordinates. Using (0, 0, 0) we see that c=0, and the desired equation is ...
x -2y +z = 0
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Additional comment
For three points, A, B, C, the vectors AB and AC will lie in the plane. Their cross product will be a vector perpendicular to the plane. That vector can be used as the coefficients of the variables in the equation for the plane. When A = (0, 0, 0), the vectors AB and AC are essentially B and C. Thus the direction vector for the plane is B×C.
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