83 Vectomnbr Parametric and Symmetric Equations of a Line in R3

7/23/2019 83 Vectomnbr Parametric and Symmetric Equations of a Line in R3

http://slidepdf.com/reader/full/83-vectomnbr-parametric-and-symmetric-equations-of-a-line-in-r3 1/2

Calculus and Vectors – How to get an A+

8.3 Vector, Parametric, and Symmetric Equations of a Line in R3

©2010 Iulia & Teodoru Gugoiu - of 2


A Vector EquationThe vector equation of the line is:

Rt ut r r ∈+= ,0

rrr

where:

OP r =r

is the position vector of a generic point

P on the line, 00 OP r =

r

is the position vector of a specific

point 0 P on the line,

ur

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 passesthrough the points )3,2,1( A and )0,1,2( − B .

If we use the direction vector )3,3,1( −−== ABur

and the point

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

Rt t r L ∈−−+= ),3,3,1()3,2,1(: r

If we use the direction vector )3,3,1(−== BAur

and the point

L B ∈− )0,1,2( , then the vector equation of the line L is:

R s sr L ∈−+−= ),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

Rt t r L ∈−+−= ),2,0,1()3,0,2(:1

r

.

R s sr L ∈−=∴ ),2,0,1(:2

r

B Specific Lines

A line is parallel to the x-axis if 0),0,0,( ≠= x x uuur

.

In this case, the line is also perpendicular to theyz-plane.

A line with 0,0),,,0( ≠≠= z y z y uuuuur

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

Rt t r ∈+−= ),0,1,0()0,2,3(r

b) passes through )4,0,1(− M and is perpendicular to the yz-

plane

Rt t r ∈+−= ),0,0,1()4,0,1(r

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

Rt bat r ∈+= ),,,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

Rt bat r ∈= ),,0,(r

. At least one of a or b is not 0 .

C Parametric Equations Let rewrite the vector equation of a line:

Rt ut r r ∈+= ,0

rrr

as:

Rt uuut z y x z y x z y x ∈+= ),,,(),,(),,( 000

The parametric equations of a line in R3 are:

Rt

tu z z

tu y y

tu x x

z

y

x

∈⎪⎩

⎪⎨

⎧

+=

+=

+=

,

0

0

0

Ex 4. Find the parametric equations of the line L that passesthrough the points )2,1,0( − A and )3,1,1( − B . Describe the line.

Rt

t z

y

t x

L

t z y x

Rt t r L

L A ABu

∈⎪⎩

⎪⎨

⎧

+=

−=

=

∴

+−=

∈+−=

∈−==

,

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.

7/23/2019 83 Vectomnbr Parametric and Symmetric Equations of a Line in R3

http://slidepdf.com/reader/full/83-vectomnbr-parametric-and-symmetric-equations-of-a-line-in-r3 2/2

Calculus and Vectors – How to get an A+


©2010 Iulia & Teodoru Gugoiu - of 2

D Symmetric Equations The parametric equations of a line may be writtenas:

Rt

tu z z

tu y y

tu x x

z

y

x

∈⎪⎩

⎪⎨

⎧

=−

=−

=−

,

0

0

0

From here, the symmetric equations of the lineare:

0,0,0

000

≠≠≠

−=

−=

−

z y x

z y x

uuu

u

z z

u

y y

u

x x

Ex 5. Convert the vector equation of the line

Rt t r L ∈−+−= ),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 y x

z y x

Rt

z

t y

t x

t z y x

Ex 6. Convert the symmetric equations for a line:

42

1

3

2 z y x=

−

+=

− to the parametric and vector

equations.

Rt t r

Rt

t z

t y

t x

t z t y

t x

t

z y x

∈−+−=∴

∈⎪⎩

⎪⎨

⎧

=

−−=

+=

∴

⎪⎩

⎪

⎨

⎧

=−=+

=−

⇒==−

+

=

−

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

,

4

21

32

421

32

42

1

3

2

r

Ex 7. For each case, find if the given point lies on the givenline.

a) )7,4,1();2,1,0()3,2,1(: −−+−= P t r L r

L P t t

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

t y

t x

L⎪⎩

⎪⎨

⎧

=

−=

+−=

⎪⎩

⎪⎨

⎧

⇒=

∉⇒∴⎭⎬⎫

−=⇒−=

=⇒+−=

true

L P t t

t t

55

11

3/2320

c) )3,3,3(;31

2

2

1: −−

−=

−=

−

+ P

z y x L

L P ∈⇒== −

−=

−=

−

+−

1113

3

1

23

2

13

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

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

Ex 7. Consider the line Rt t r L ∈−−+−= ),3,2,1()3,2,3(: r

. Find

the intersection points between this line and the coordinatesaxes and planes.

axis x L B x

plane xy L B xy

y xt z

plane xz L B xz

z xt y

plane yz L A yz

z yt x

t z

t y

t x

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

83 Vectomnbr Parametric and Symmetric Equations of a Line in R3

Documents

Transcript of 83 Vectomnbr Parametric and Symmetric Equations of a Line in R3