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Definition 1.6
Let
and
be two vectors in
.
The
dot product, or
scalar product of
and
is the real number
Proposition 1.8
Two vectors
and
are orthogonal if, and only if,
.
Example 1.9
If
and
,
then
.
Thus
.
Definition 1.10
Let
be a plane whose cartesian equation is
ax+
by+
cz+
d=0. The vector
is called
a normal vector for
.
Figure 5:
A vector normal to a plane.

Example 1.11
The vector
is normal to the plane
whose equation is
x + 2
y 3
z +5 =0.
Proposition 1.12
Let
and
be two planes with respective normal vectors
and
.
The two planes
and
are orthogonal if, and only if, the vectors
and
are orthogonal.
Example 1.13
Let
and
.
These planes have
respective normal vectors
and
.
We have:
,
thus
.
Proposition 1.14
A line
and a plane
are perpendicular if, and only if, a direction vector for
is a normal vector for
.
Example 1.15
Let
be the line given by the parametric equations
and let
be the plane whose cartesian equation is
2
x+4
y3
z1=0. Then
and
are perpendicular.
Example 1.16
If
is defined by the equation
3
x
y+5
z6=0, then the line
through the point
A(5,2,7) and perpendicular to
has the following parametric equations:
Next: Cross product.
Up: Analytic geometry in the
Previous: Planes and lines.
Noah DanaPicard
20010530