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

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