Chapter 11 - Vectors and the Geometry of Space - 11.5 Exercises - Page 791: 42

\[-x-2y+z-1=0\]

\[\begin{align} & \text{Points }\underbrace{\left( 3,-1,2 \right)}_{\left( {{x}_{1}},{{y}_{1}},{{z}_{1}} \right)},\left( 2,1,5 \right),\left( 1,-2,-2 \right) \\ & \text{Let }\mathbf{u}\text{ be the vector from }\left( 3,-1,2 \right)\text{ to }\left( 2,1,5 \right) \\ & \mathbf{u}=\left\langle 2-3,1+1,5-2 \right\rangle \\ & \mathbf{u}=\left\langle -1,2,3 \right\rangle \\ & \text{Let }\mathbf{v}\text{ be the vector from }\left( 3,-1,2 \right)\text{ to }\left( 1,-2,-2 \right) \\ & \mathbf{v}=\left\langle 1-3,-2+1,-2-2 \right\rangle \\ & \mathbf{v}=\left\langle -2,-1,-4 \right\rangle \\ & \mathbf{n}=\mathbf{u}\times \mathbf{v}=\left| \begin{matrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ -1 & 2 & 3 \\ -2 & -1 & -4 \\ \end{matrix} \right| \\ & \mathbf{n}=\mathbf{u}\times \mathbf{v}=\left| \begin{matrix} 2 & 3 \\ -1 & -4 \\ \end{matrix} \right|\mathbf{i}-\left| \begin{matrix} -1 & 3 \\ -2 & -4 \\ \end{matrix} \right|\mathbf{j}+\left| \begin{matrix} -1 & 2 \\ -2 & -1 \\ \end{matrix} \right|\mathbf{k} \\ & \mathbf{n}=-5\mathbf{i}-10\mathbf{j}+5\mathbf{k} \\ & \mathbf{n}=\left\langle a,b,c \right\rangle \\ & a=-5,\text{ }b=-10,\text{ }c=5 \\ & \text{The equation of the plane is given by} \\ & a\left( x-{{x}_{1}} \right)+b\left( y-{{y}_{1}} \right)+c\left( z-{{z}_{1}} \right)=0 \\ & \text{Where }\underbrace{\left( 3,-1,2 \right)}_{\left( {{x}_{1}},{{y}_{1}},{{z}_{1}} \right)},\text{ then} \\ & -5\left( x-3 \right)-10\left( y+1 \right)+5\left( z-2 \right)=0 \\ & -5x+15-10y-10+5z-10=0 \\ & -5x-10y+5z-5=0 \\ & -x-2y+z-1=0 \\ \end{align}\]