Planes

Pre-requisites: Dot Product

Planes play the same fundamental role in vector calculus as lines do in single variable calculus, for the simple reason that functions like f(x,y) don't have tangent lines, they have tangent planes.

We have already seen several examples of planes; for example, the equation z = 0 implicitly defines the x-y plane. Remember that in an implicit description, any unmentioned variables are free to do whatever they want: the equation y = m x + b which used to describe a line in the plane now describes a plane in R3 because z can be anything.

In general, a plane in R3 is implicitly described by the equation

A x + B y + C z + D = 0
It is frequently more useful to rewrite this equation as
A ( x - x0 ) + B ( y - y0 ) + C ( z - z0 ) = 0
where ( x0, y0, z0 ) is some point on the plane, or even as
(A,B,C) · ( x - x0, y - y0, z - z0 ) = 0

This last equation allows us to see (A,B,C) as a normal vector to the plane: a vector which is perpendicular to every vector lying in the plane. (A vector lies in a plane if it is the displacement vector of two points that lie in the plane). Thus, one way to find the equation of a plane is to find a normal vector n and a point P, and then the plane will have the equation

n · ( (x,y,z) - OP ) = 0
Because there are lots of different choices for P, and since if n is a normal vector, so is cn, we see that there are lots of different equations for the same plane.

Another way to specify a plane is to give three points that lie on that plane. If we have the three points' cartesian coordinates, we can find the equation of the plane they lie on by solving a system of linear equations. For example, given the points (6,1,4), (-1,-5,-2), and (-1,1,-1), we solve the equations 6 A + B + 4 C + D = 0, -A - 5 B - 2 C + D = 0, -A + B - C + D = 0, to get that A = (30/7) B, C = -6 B, and D = -(19/7) B. Any choice of B works (to repeat, there are many different equations for the same plane), so we can choose, say, B = 7 and get the equation 30 x + 7 y - 42 z - 19 = 0.

Finally, you might be wondering what the parametric description of a plane looks like. A plane is a surface, and like all surfaces, requires two parameters to be described parametrically. When converting the plane x + 3y - 7z + 11 = 0 to a parametric description, it is easiest to just let two of the variables -- say y and z -- be the parameters, and solve for x, yielding the equation

P(u,v) = ( -11 - 3u + 7v, u, v ) = (-11,0,0) + (-3,1,0) u + (7,0,1)v
In general, a plane will have a parametric description of the form
P(r,s) = OP + v r + w s
where P is a point on the plane and v and w are two vectors lying on the plane.

Exercises:

  1. What is the equation of the yz plane?
  2. Give a normal vector to the plane x - 2x + 3y = 4.
  3. Find the plane going through the origin that has (1,2,3) as its normal vector.
  4. Find the plane going through (1,2,3), (4,5,6) and (7,0,-2).
  5. Find the plane containing the point (1,1,-3) and the line l(t) = (3,1/2,-1) t + (0,0,3).
  6. Show that if a plane contains two points on a line, then it contains the entire line.
  7. Find the plane containing the two intersecting lines l1(t) = (1,2,3) t + (4,5,6) and l1(t) = (-2,2,9) t + (4,5,6).
  8. Find the plane containing the two parallel lines l1(t) = (1,2,3) t + (4,5,6) and l1(t) = (2,4,6) t + (0,0,3).
  9. Convert the plane x - 2x + 3y = 4 to parametric form.
  10. Convert the plane P(r,s) = (1,2,3) + (4,5,6) r + (7,8,9) s to implicit form.
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Last modified 1 July 1997