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.


About Alex Nelson

I like math. I like programming. Most of all, I love puzzles.
This entry was posted in Calculus, Cross Product, Geometry, Vector Calculus and tagged . Bookmark the permalink.

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