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ET 12.5; M 13.5. Lines and Planes
Internal descriptions.
To specify a line, give the
position,
~
r
0
say, of some point in the line and give a
nonzero vector,
~v
say, running along the line. Then
the position
~
r
of a point in the line satisﬁes the
vector
equations
~
r

~
r
0
=
t~v,
and
~
r
=
~
r
0
+
t~v.
(
*
)
Having two distinct points in the line also leads to
such equations. When the
parameter
t
is time, the
vector
~v
is velocity.
In
R
3
, the vector equation (
*
) corresponds to
three scalar
parametric equations.
Solve for
t
in
those equations to get
symmetric equations
.
The idea in (
*
) works for a plane in
R
3
, but
we need two nonparallel vectors lying in the plane.
Then
~
r
=
~
r
0
+
s~u
+
t~v
(
†
)
with
two
parameters. Such equations also come from
having
three
nonaligned points in the plane.
1
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View Full DocumentET 12.5; M 13.5. Lines and Planes
External descriptions.
In equation (
†
), the two
vectors
~u
and
~v
implicitly determine how the plane
is tilted. In
R
3
, it is simpler to specify one nonzero
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 Spring '03
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