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12.5 Equations of Lines and Planes
A line
L
is determined by a point
on L and the direction of L. The direction of a
line is described by a vector.
(
0000
,,
Pxyz
)
 vector parallel to L
point on L
 posi
epresentation vectors)

riangle Law for vector addition gives
a
r
lar
such that
v
.
v
r
v
(, ,)
xyz
 an arbitrary
P
rr
tion vectors of P
0
and P (r
0
,
r
vector with representation
P
uuur
a
0
P
T
0
=+
a
r
and
v
r
are parallel and there is a sca
t
at
=
0
t
rr r
is the
vector equation of L.
ach value of the parameter
gives the position vector
of a point on L.
corresponds to points on L on one side of P
0
.
of P
0
.
: vector that gives the direction of L
E
t
r
r
0
>
corresponds to points on L on the other side
t
0
t
<
v
r
abc
=
is the direction of L in co
r
v
tv
ta tb tc
=
r
,
, ,
r x
y
z
=
mponent form from which we can get
.
00
0
rx
y
z
=
0
The
vector equation
of the line
r
comes
0
rrt
v
be
000
if the corresponding components are equal.
P(x,y,z)
r
x
y
z
L
v
P
0
(x
0
,y
0
,z
0
)
r
0
t=0
t>0
t<0
r
0
a
,
,
x y z
x
ta y
tb z
tc
+
+
Two vectors are equal if and only
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This note was uploaded on 01/20/2012 for the course MATH 2057 taught by Professor Estrada during the Fall '08 term at LSU.
 Fall '08
 Estrada
 Equations

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