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Answer:
The equation of the plane is represented by [tex]\frac{1}{18}\cdot x + \frac{1}{18}\cdot y + \frac{1}{18}\cdot z = 1[/tex].
Step-by-step explanation:
Algebraically speaking, a plane can be represented by following vectorial product:
[tex](a,b, c)\,\bullet\,(x,y,z) = 1[/tex] (1)
Where:
[tex]a[/tex], [tex]b[/tex], [tex]c[/tex] - Plane coeffcients.
[tex]x[/tex], [tex]y[/tex], [tex]z[/tex] - Coordinates.
We need three distinct points to determine all coefficients. If we know that [tex](x_{1},y_{1},z_{1}) = (0,9,9)[/tex], [tex](x_{2},y_{2}, z_{2}) = (9,0,9)[/tex] and [tex](x_{3},y_{3},z_{3}) = (9,9,0)[/tex], the system of equations to be solved is:
[tex]9\cdot b + 9\cdot c = 1[/tex] (1)
[tex]9\cdot a + 9\cdot c = 1[/tex] (2)
[tex]9\cdot a + 9\cdot b = 1[/tex] (3)
The solution of this system is [tex]a = \frac{1}{18}[/tex], [tex]b = \frac{1}{18}[/tex], [tex]c = \frac{1}{18}[/tex].
Hence, the equation of the plane is represented by [tex]\frac{1}{18}\cdot x + \frac{1}{18}\cdot y + \frac{1}{18}\cdot z = 1[/tex].