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Step 1 Our surface S is part of the plane containing the three points (1,0,0),(0,9,0), and (0,0,9). The equation of this plane is z=

\newlineOur surface S S is part of the plane containing the three points (1,0,0),(0,9,0) (1,0,0),(0,9,0) , and (0,0,9) (0,0,9) . The equation of this plane is\newlinez= z=

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Q. \newlineOur surface S S is part of the plane containing the three points (1,0,0),(0,9,0) (1,0,0),(0,9,0) , and (0,0,9) (0,0,9) . The equation of this plane is\newlinez= z=
  1. Find Normal Vector: Step 11: Identify the normal vector to the plane using the given points.\newlineTo find the normal vector, we use two vectors in the plane and find their cross product. The vectors can be formed using the given points: vector AB from (1,0,0)(1,0,0) to (0,9,0)(0,9,0) and vector AC from (1,0,0)(1,0,0) to (0,0,9)(0,0,9).\newlineVector AB = (0−1,9−0,0−0)(0-1, 9-0, 0-0) = (−1,9,0)(-1, 9, 0)\newlineVector AC = (0−1,0−0,9−0)(0-1, 0-0, 9-0) = (−1,0,9)(-1, 0, 9)\newlineCross product of AB and AC = \begin{vmatrix}i & j & k\-1 & 9 & 0\-1 & 0 & 9\end{vmatrix}\newline= i(9⋅9−0⋅0)−j(−1⋅9−0⋅(−1))+k(−1⋅0−9⋅(−1))i(9\cdot9 - 0\cdot0) - j(-1\cdot9 - 0\cdot(-1)) + k(-1\cdot0 - 9\cdot(-1))\newline= (0,9,0)(0,9,0)00\newlineNormal vector = (0,9,0)(0,9,0)11
  2. Write Plane Equation: Step 22: Write the equation of the plane using the normal vector and a point on the plane.\newlineThe general form of the plane equation is Ax+By+Cz=DAx + By + Cz = D, where (A,B,C)(A, B, C) is the normal vector and (x,y,z)(x, y, z) is any point on the plane.\newlineUsing the normal vector (81,9,9)(81, 9, 9) and point (1,0,0)(1,0,0), substitute into the plane equation:\newline81×1+9×0+9×0=D81\times1 + 9\times0 + 9\times0 = D\newline81=D81 = D\newlineSo, the equation of the plane is 81x+9y+9z=8181x + 9y + 9z = 81
  3. Simplify Plane Equation: Step 33: Simplify the equation of the plane.\newlineDivide the entire equation by 99 to simplify:\newline819x+99y+99z=819\frac{81}{9}x + \frac{9}{9}y + \frac{9}{9}z = \frac{81}{9}\newline9x+y+z=99x + y + z = 9\newlineThus, the simplified equation of the plane is z=9−9x−yz = 9 - 9x - y

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