Calculus and Vectors – How to get an A+

8.2 Cartesian Equation of a Line ©2010 Iulia & Teodoru Gugoiu - Page 1 of 2

8.2 Cartesian Equation of a Line A Symmetric Equation The parametric equations of a line in R2:

Rttuyytuxx

y

x ∈⎩⎨⎧

+=+=

0

0

may be written as:

Rttuyy

uxx

yx∈=

−=

− ,00

The symmetric equation of the line is (if exists):

yx uyy

uxx 00 −

=−

Note. The symmetric equation does exist if

0≠xu and 0≠yu .

Ex 1. Convert the vector or parametric equations to the symmetric equation (if exists).

a) Rttytx

∈⎩⎨⎧

−=+−=523

b) Rttr ∈−+= ),2,1()1,0(

r c) Rttr ∈+−= ),0,2()3,2(

r Ex 2. Convert the symmetric equation of a line L :

23

12 −=

−+ yx to the vector equation.

B Normal Equation Let consider a line L that passes through the specific point ),( 000 yxP and has the direction vector ),( yx uuu =

r .

The vectors ),(),( BAuun xy =−=

r or

),(),( BAuun xy =−=r are perpendicular to the vector

ur and so they are perpendicular to the line L . These are called normal vectors to the line L . Let ),( yxP be a generic point on the line L . So:

0)(0||

0

000

=⋅−=⋅⇒⊥⇒

nrrnPPnPPuPP

rrr

rrr

The normal equation of a line is given by: 0)( 0 =⋅− nrr

rrr

C Cartesian Equation The normal equation can be written as:

00

00

00

0

00

0),(),(),(),(0

ByAxCwhereCByAxByAxByAx

BAyxBAyxnrnr

−−==++=−−+

=⋅−⋅=⋅−⋅

rrrr

The Cartesian equation of a line is given by:

0=++ CByAx where

),( BAn =r is a normal vector and the constant C depends on

a specific point of the line.

Calculus and Vectors – How to get an A+

8.2 Cartesian Equation of a Line ©2010 Iulia & Teodoru Gugoiu - Page 2 of 2

Ex 3. Convert the vector equation of the line RttrL ∈−+= ),2,3()2,1(:

r to the Cartesian equation.

Ex 4. Convert the Cartesian equation to the parametric equations and then to the vector equation.

032: =++− yxL

D Slope y-intercept Equation Let solve the symmetric equation of a line:

Rttuyy

uxx

yx∈=

−=

− ,00

for y :

00

00

xuu

yxuu

y

uxxuyy

x

y

x

y

xy

−+=

−=−

The slope y-intercept equation of a line in R2 is given by:

x

y

uu

m

nmxy

=

+=

where m is the slope and n is the y-intercept which depends on a specific point of the line.

Ex 5. Convert the vector equation to the slope y-intercept equation: )2,1()3,2(: −+−= trL

r Ex 6. Convert the slope y-intercept equation to the vector equation.

432: +−= xyL .

E Angle between two Lines The angle between two lines is determined by the angle between the direction vectors:

||||||||cos

21

21

uuuurr

rr⋅

=θ

Note. There are two pairs of equal angles between two lines (see the figure below).

Note: °=+ 18021 θθ

Ex 7. Find the acute angle between each pair of lines.

a) 21

32:;),3,1()1,0(: 21 −

+=

−∈−+=

yxLRttrLr

b) 231:;0632: 21 +−==+− xyLyxL

Reading: Nelson Textbook, Pages 435-442 Homework: Nelson Textbook: Page 443 #1, 3, 4, 5, 6, 7, 8, 10ab, 11, 14