3-Dimensional planes

3-Dimensional planes

Lessons

Notes:

Equation of a Plane
Couple sections ago, we saw that the equation of plane can be expressed as ax+by+cz=dax+by+cz=d. However, this equation does not give much information. So suppose we have the following graph:
plane graph
Where r\vec{r} and r0\vec{r_0} are position vectors for points PP and P0P_0 respectively, and n\vec{n} is a normal vector that is orthogonal (perpendicular) to the plane.
Since rr0\vec{r} - \vec{r_0} is on the plane, then n\vec{n} is orthogonal to rr0\vec{r} - \vec{r_0}. In other words, their dot products should give 0.
So,

(rr0)n=0(<x,y,z><x0,y0,z0>)<a,b,c>=0(\vec{r} - \vec{r_0}) \cdot \vec{n} = 0 \to (\lt x,y,z\gt - \lt x_0,y_0, z_0\gt) \cdot \lt a,b,c\gt = 0
<xx0,yy0,zz0><a,b,c>=0\to \lt x-x_0 , y-y_0, z-z_0\gt \cdot \lt a,b,c\gt = 0
a(xx0)+b(yy0)+c(zz0)=0\to a(x-x_0) + b(y-y_0) + c(z-z_0) = 0

Which is formula for the equation of a plane.

The key to finding the equation of a plane is finding two things:
  1. The normal vector (orthogonal to the plane)
  2. A point on the plane.
Then you can just plug those into the formula to get the equation!
  • Introduction
    3-Dimensional Planes Overview:
    a)
    Equation of a Plane
    • How do we get the formula for the equation?
    • a(xx0)+b(yy0)+c(zz0)=0a(x-x_0) + b(y-y_0) + c(z-z_0) = 0
    • What we need for the formula

    b)
    Finding a Plane with a Parallel Plane & 1 point
    • Get the Normal Vector
    • Plug into the formula

    c)
    Finding the Equation of a Plane with 3 points
    • Creating 2 vectors
    • Using the Cross Product = Normal Vector
    • Plug into the formula


  • 1.
    Finding the Equation of a Plane
    Find the equation of the plane which contains the points (1,3,0)(1, 3, 0), (2,7,6)(-2, 7, 6) and (1,0,1)(1, 0, 1).

  • 2.
    Find the equation of the plane which contains the point (0,2,1)(0, -2, 1) and is orthogonal to the line <1+2t,t,0><1+2t, t, 0>.

  • 3.
    Are the Two Planes Parallel, Orthogonal or Neither?
    Determine whether the two planes 2x+4y+6z=82x+4y+6z=8 and x+2y+3z=1x+2y+3z=1 are parallel, orthogonal, or neither.

  • 4.
    Determine whether the two planes 3x+y+8z=4-3x+y+8z=4 and 2x+6y=12x+6y=1 are parallel, orthogonal, or neither.

  • 5.
    Intersection of a Plane and a Line
    Determine whether the plane 3x+5y+z=2-3x+5y+z=2 and line r(t)=<2+3t,5t,1t>r(t)=\lt2+3t, -5t, 1-t\gt intersect.