Chapter 11 - Vectors and the Geometry of Space - Review Exercises - Page 812: 47

\[27x+4y+32z=-33\]

\[\begin{align} & \text{Points }\underbrace{\left( -3,-4,2 \right)}_{\left( {{x}_{1}},{{y}_{1}},{{z}_{1}} \right)},\left( -3,4,1 \right),\left( 1,1,-2 \right) \\ & \text{Let }\mathbf{u}\text{ be the vector from }\left( -3,-4,2 \right)\text{ to }\left( -3,4,1 \right) \\ & \mathbf{u}=\left\langle -3+3,4-\left( -4 \right),1-2 \right\rangle \\ & \mathbf{u}=\left\langle 0,8,-1 \right\rangle \\ & \text{Let }\mathbf{v}\text{ be the vector from }\left( -3,-4,2 \right)\text{ to }\left( 1,1,-2 \right) \\ & \mathbf{v}=\left\langle 1+3,1+4,-2-2 \right\rangle \\ & \mathbf{v}=\left\langle 4,5,-4 \right\rangle \\ & \mathbf{n}=\mathbf{u}\times \mathbf{v}=\left| \begin{matrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ 0 & 8 & -1 \\ 4 & 5 & -4 \\ \end{matrix} \right| \\ & \mathbf{n}=\mathbf{u}\times \mathbf{v}=\left| \begin{matrix} 8 & -1 \\ 5 & -4 \\ \end{matrix} \right|\mathbf{i}-\left| \begin{matrix} 0 & -1 \\ 4 & -4 \\ \end{matrix} \right|\mathbf{j}+\left| \begin{matrix} 0 & 8 \\ 4 & 5 \\ \end{matrix} \right|\mathbf{k} \\ & \mathbf{n}=-27\mathbf{i}-4\mathbf{j}-32\mathbf{k} \\ & \mathbf{n}=\left\langle a,b,c \right\rangle \\ & a=-27,\text{ }b=-4,\text{ }c=-32 \\ & \text{The equation of the plane is given by} \\ & a\left( x-{{x}_{1}} \right)+b\left( y-{{y}_{1}} \right)+c\left( z-{{z}_{1}} \right)=0 \\ & \text{Where }\underbrace{\left( -3,-4,2 \right)}_{\left( {{x}_{1}},{{y}_{1}},{{z}_{1}} \right)},\text{ then} \\ & -27\left( x+3 \right)-4\left( y+4 \right)-32\left( z-2 \right)=0 \\ & -27x-81-4y-16-32z+64=0 \\ & 27x+4y+32z=-33 \\ \end{align}\]