Vector, Parametric, and Symmetric Equations of a Line

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Oct 13, 2013 (3 years and 7 months ago)

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Calculus and Vectors – How to get an A+

8.3 Vector, Parametric, and Symmetric Equations of a Line in R
3

©2010 Iulia & Teodoru Gugoiu - Page 1 of 2
8.3 Vector, Parametric, and Symmetric Equations of a Line in R
3


A Vector Equation
The vector equation of the line is:
Rtutrr ∈+=,
0
r
r
r

where:

OPr =
r
is the position vector of a generic point
P
on the line,

00
OPr =
r
is the position vector of a specific
point
0
P
on the line,

u
r
is a vector parallel to the line called the
direction vector of the line, and

t
is a real number corresponding to the
generic point
P
.

Ex 1. Find two vector equations of the line
L
that passes
through the points
)3,2,1(A
and
)0,1,2( −B
.

If we use the direction vector
)3,3,1( −−== ABu
r
and the point
LA

)3,2,1(
, then the vector equation of the line
L
is:
RttrL



+
=
),3,3,1()3,2,1(:
r


If we use the direction vector
)3,3,1(−== BAu
r
and the point
LB


)0,1,2(
, then the vector equation of the line
L
is:
RssrL


+

=
),3,3,1()0,1,2(:
r


Ex 2. Find the vector equation of a line
2
L
that passes
through the origin and is parallel to the line
RttrL


+

=
),2,0,1()3,0,2(:
1
r
.

RssrL


=

),2,0,1(:
2
r

B Specific Lines
A line is parallel to the x-axis if
0),0,0,(

=
xx
uuu
r
.
In this case, the line is also perpendicular to the
yz-plane.

A line with
0,0),,,0( ≠≠=
zyzy
uuuuu
r
is parallel to
the yz-plane.


Ex 3. Find the vector equation of a line that:

a) passes through
)0,2,3(

A
and is parallel to the y-axis
Rttr

+

=
),0,1,0()0,2,3(
r


b) passes through
)4,0,1(

M
and is perpendicular to the yz-
plane
Rttr

+

=
),0,0,1()4,0,1(
r


c) passes through
)0,0,3(P
and is perpendicular to the x-axis
Rtbatr

+
=
),,,0()0,0,3(
r
. At least one of
a
or
b
is not
0
.

d) passes through the origin and is parallel to the xz-plane
Rtbatr

=
),,0,(
r
. At least one of
a
or
b
is not
0
.
C Parametric Equations
Let rewrite the vector equation of a line:
Rtutrr ∈+=,
0
r
r
r

as:
Rtuuutzyxzyx
zyx
∈+= ),,,(),,(),,(
000

The parametric equations of a line in R
3
are:
Rt
tuzz
tuyy
tuxx
z
y
x






+=
+=
+=
,
0
0
0

Ex 4. Find the parametric equations of the line
L
that passes
through the points
)2,1,0(

A
and
)3,1,1( −B
. Describe the line.

Rt
tz
y
tx
L
tzyx
RttrL
LAABu






+=
−=
=

+−=
∈+−=
∈−==
,
2
1
)1,0,1()2,1,0(),,(
),1,0,1()2,1,0(:
)2,1,0();1,0,1(
r
r

The line is parallel to the xz-plane.
Calculus and Vectors – How to get an A+

8.3 Vector, Parametric, and Symmetric Equations of a Line in R
3

©2010 Iulia & Teodoru Gugoiu - Page 2 of 2
D Symmetric Equations
The parametric equations of a line may be written
as:
Rt
tuzz
tuyy
tuxx
z
y
x






=−
=−
=−
,
0
0
0

From here, the symmetric equations of the line
are:
0,0,0
000
≠≠≠

=

=

zyx
zyx
uuu
u
zz
u
yy
u
xx

Ex 5. Convert the vector equation of the line
RttrL


+

=
),0,2,1()3,1,0(:
r
to the parametric and
symmetric equations.

3,
2
1
1
0
3
2
1
1
,
3
21
)0,2,1()3,1,0(),,(
−=

=


+
=

=







−=
+=
−=


+

=
z
yx
zyx
Rt
z
ty
tx
tzyx

Ex 6. Convert the symmetric equations for a line:
42
1
3
2 zyx
=

+
=

to the parametric and vector
equations.
Rttr
Rt
tz
ty
tx
tz
ty
tx
t
zyx
∈−+−=∴






=
−−=
+=






=
−=+
=−
⇒==

+
=

),4,2,3()0,1,2(
,
4
21
32
4
21
32
42
1
3
2
r

Ex 7. For each case, find if the given point lies on the given
line.
a)
)7,4,1();2,1,0()3,2,1(:−

+

=
PtrL
r

LPtt
t
t
∈⇒∴=⇒−=−
−=−−−

+

=

2)2,1,0()4,2,0(
)2,1,0()3,2,1()7,4,1(
)2,1,0()3,2,1()7,4,1(

b)
)5,1,0(;
5
32
:P
z
ty
tx
L





=
−=
+−=






⇒=
∉⇒∴



−=⇒−=
=⇒+−=
true
LP
tt
tt
55
11
3/2320

c)
)3,3,3(;
31
2
2
1
:−−

=

=

+
P
zyx
L

LP∈⇒==


=

=

+

111
3
3
1
23
2
13

E Intersections
A line intersects the x-axis when
0== zy
.



A line intersects the xy-plane when
0=z
.



Ex 7. Consider the line
RttrL

−−+

=
),3,2,1()3,2,3(:
r
. Find
the intersection points between this line and the coordinates
axes and planes.

axisxLBx
planexyLBxy
yxtz
planexzLBxz
zxty
planeyzLAyz
zytx
tz
ty
tx
L
−∩==−∴
−∩==−∴
=+−==−=⇒=⇒=
−∩==−∴
=−==−=⇒=⇒=
−∩=−=−∴
−=−==+−=⇒=⇒=





−=
+−=
−=
)0,0,2(int
)0,0,2(int
0)1(22,21310
)0,0,2(int
0)1(33,21310
)6,4,0(int
6)3(33,4)3(2230
33
22
3
:

Note that y-intercept and z-intercept do not exist.

Reading: Nelson Textbook, Pages 445-448
Homework: Nelson Textbook: Page 449 #1abc, 5acf, 6, 8, 9, 12, 13, 14