Planes in the space
An equation of a plane in space can be obtained from a point in the plane and a vector normal (perpendicular) to the plane.
Consider the plane containing the point P(x1,y1,z1) having a nonzero normal vector n=<a,b,c>, as shown in the next figure:
This plane consists of all points Q(x,y,z) for which vector is orthogonal to n.
Using the dot product, you can write the following:
<a, b, c>·<x-x1, y-y1, z-z1>=0
a(x-x1) + b(y-y1) + c(z-z1) = 0The third equation of the plane is said to be in standard form.
STANDARD EQUATION OF A PLANE IN SPACE.
The plane containing the point (x1,y1,z1) and having normal vector n = <a,b,c> can be represented by the standard form of the equation of a plane:
By regruping terms, you obtain the general form of the equation of a plane in space:
Given the general form of the equation of a plane, it is easy to find a normal vector to the plane. Simply use the coefficients of x, y and z and write n = <a,b,c>
Find the general equation of the plane containing the points (2,1,1), (0,4,1), and (-2,1,4)
The component forms of u and v are:
u = <0 - 2, 4 - 1, 1 - 1> = <-2, 3, 0>
v = <-2 - 2, 1 - 1, 4 - 1> = <-4, 0, 3>
and it follows that
n = u x v =
=9i + 6j +12k
is normal to the given plane. Using the direction numbers for n and the point (x1,y1,z1)=(2,1,1), you can determine an equation of the plane to be
PARAMETRIC EQUATIONS OF PLANES IN SPACE.
Supose that u = <u1,u2,u3> and v = <v1,v2,v3> are nonparallel vectors that lie in a plane V. Let P0=(x0,y0.z0) be any fixed point that lies on V. We can think of P0 as an "origin" in the plane. Any point P=(x,y,z) on the plane can be reached from P0 by adding a scalar multiple of u and then a scalar multiple of v, as illustrate in the following picture:
The position vector can be obtained by vector addition as:
= + u + v
If we write out this vector equation in coordinates, then we get three scalar parametric equations for the plane:
The parameters and may be thought of as coordinates in the plane V: Each point of V is determined by specifying the values of and .