## Constructing Planes

1. We can use vectors and the cross product to construct planes quite easily.

2. Determining a Plane From a Point and Normal Vector. Given a vector
(1)$\displaystyle \vec{N} = A\widehat{\textbf{\i}}+b\widehat{\textbf{\j}}+C\widehat{\textbf{k}}$
and a point ${P_{0}=(x_{0},y_{0},z_{0})}$, there exists a unique plane which is perpindicular to ${\vec{N}}$ and contains ${P_{0}}$.

How? Well, let ${P}$ be an arbitrary point on the plane. Then the vector ${\overrightarrow{P_{0}P}}$ would be parallel to the plane. Being parallel to the plane implies its orthogonal to ${\vec{N}}$:
(2)$\displaystyle \overrightarrow{P_{0}P}\cdot\vec{N}=0.$
This gives us an equation
(3)\displaystyle \begin{aligned} \overrightarrow{P_{0}P}\cdot\vec{N}&= \bigl( (x-x_{0})\widehat{\textbf{\i}}+(y-y_{0})\widehat{\textbf{\j}}+(z-z_{0})\widehat{\textbf{k}} \bigr)\cdot(A\widehat{\textbf{\i}}+b\widehat{\textbf{\j}}-C\widehat{\textbf{k}}) \\ &=A(x-x_{0})+B(y-y_{0})+C(z-z_{0})=0. \end{aligned}
So any point ${(x,y,z)}$ lies on the plane if it satisfies this equation.

3. Determine a Plane from Three Points. Given three points ${A}$, ${B}$, ${C}$, find a plane containing these points.

We construct the vectors ${\overrightarrow{AB}}$ and ${\overrightarrow{AC}}$. Take the cross product, which produces the normal vector
(4)$\displaystyle \vec{N} = \overrightarrow{AB}\times\overrightarrow{AC}.$
If we write ${A=(x_{0},y_{0},z_{0})}$ and ${\vec{N}=\langle N_{1},N_{2},N_{3}\rangle}$, then
(5)$\displaystyle N_{1}(x-x_{0})+N_{2}(y-y_{0})+N_{3}(z-z_{0}) = 0$
describes the plane. (It follows from our last construction of the plane.)

Exercise 1 [Math.SE]. Determine the plane containing point $P(-5,2,3)$ and going through the intersection line of the planes $2x+y+5z=31$ and $-4x+5y+4z=50$.

Exercise 2 [Math.SE]. Let $\ell$ be the line given by $x=3-t$, $y=2+t$, $z=-4+2t$. Then $\ell$ intersects the plane $3x-2y+z=1$ at the point $P=(3,2,-4)$. Find parametric equations for the line through $P$ which lies on plane and is perpendicular to $\ell$.

Exercise 3 [Math.SE]. Find normal form equation of plane containing $A(-2,1,5)$ and Line: $(2-\mu)\widehat{\textbf{\i}}+(3+\mu)\widehat{\textbf{\j}}+(4+\mu)\widehat{\textbf{k}}$ where $\mu\in\mathbb{R}$ is some parameter.