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 xaxis if
0),0,0,(
≠
=
xx
uuu
r
.
In this case, the line is also perpendicular to the
yzplane.
A line with
0,0),,,0( ≠≠=
zyzy
uuuuu
r
is parallel to
the yzplane.
Ex 3. Find the vector equation of a line that:
a) passes through
)0,2,3(
−
A
and is parallel to the yaxis
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 xaxis
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 xzplane
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 xzplane.
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 xaxis when
0== zy
.
A line intersects the xyplane 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 yintercept and zintercept do not exist.
Reading: Nelson Textbook, Pages 445448
Homework: Nelson Textbook: Page 449 #1abc, 5acf, 6, 8, 9, 12, 13, 14
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