Section 3.5
Lines and Planes in 3-Space
POINT-NORMAL EQUATION OF A
PLANE
Let n = (a, b, c) ≠ 0 be a vector normal (perpendicular) to
the plane containing the point P0(x0, y0, z0). For any point
P(x,y, z) in the plane, the vector
P0 P  ( x  x0 , y  y0 , z  z0 )
is orthogonal to n.
That is,
n  P0 P  0
Hence, a( x  x0 )  b( y  y0 )  c( z  z0 )  0
This is called the point-normal form of a plane.
GENERAL FORM OF THE
EQUATION OF A PLANE
Theorem 3.5.1: If a, b, c, and d are constants
and a, b, and c are not all zero, then the graph of
the equation
ax + by + cz + d = 0
is a plane having the vector n = (a, b, c) as a
normal.
This equation is called the general form of the
equation of a plane.
GEOMETRIC INTERPRETATION
OF A SYSTEM OF EQUATIONS
Recall the solution to a system of linear equations with two variables and two
equation corresponds to the intersection of two lines. Similarly, the solution of
a system of three equations with three variables corresponds to the intersection
of three planes.
VECTOR FORM OF EQUATION
OF A PLANE
Let P(x, y, z) be any point in a plane and let P0(x0,
y0, z0) be a specific point in the plane. Let r0 be the
vector from the origin to P0(x0, y0, z0), r be the
vector from the origin to P(x, y, z), and n = (a, b, c)
be a vector normal to the plane. Then P0 P  r  r0 ,
so the general equation of the plane can be rewritten
as
n · (r − r0) = 0.
This is called the vector form of the equation of a
plane.
LINES IN 3-SPACE
Let v = (a, b, c) ≠ 0 be a vector parallel to the line l in 3sapce, and line l contains the point P0(x0, y0, z0). For any point
P(x, y, z) on the line l, the vector
P0 P  ( x  x0 , y  y0 , z  z0 )
is parallel to v.
That is, for some scalar t,
P0 P  t v .
Hence, (x − x0, y − y0, z − z0) = (ta, tb, tc). That is,
x = x0 + ta, y = y0 + tb, z = z0 + tc,
−∞ < t < ∞ .
These equations are called parametric equations for the line l.
VECTOR FORM OF THE
EQUATION OF A LINE
Let r = (x, y, z) be the vector from the origin to the point P(x,
y, z), let r0 = (x0, y0, z0) be the vector from the origin to the
point P0 (x0, y0, z0), and let v = (a, b, c) be the vector parallel
to the line. Then P0 P  r  r0 and the equation of the line
can be written as
r − r0 = tv
Taking into account the range of t-values, this can be written
as
r = r0 + tv
(−∞ < t < ∞)
This is called the vector form of the equation of a line in 3space
DISTANCE BETWEEN AND
POINT AND A PLANE
Theorem 3.5.2: The distance D between a point
P0(x0, y0, z0) and the plane ax + by + cz + d = 0 is
D
| ax0  by0  cz0  d |
a b c
2
2
2