h82_Cartesian_Equation_of_a_Line.pdf
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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 R 2 : R t tu y y tu x x y x ∈ ⎩ ⎨ ⎧ + = + = 0 0 may be written as: R t t u y y u x x y x ∈ = − = − , 0 0 The symmetric equation of the line is (if exists): y x u y y u x x 0 0 − = − Note. The symmetric equation does exist if 0 ≠ x u and 0 ≠ y u . Ex 1. Convert the vector or parametric equations to the symmetric equation (if exists). a) R t t y t x ∈ ⎩ ⎨ ⎧ − = + − = 5 2 3 b) R t t r ∈ − + = ), 2 , 1 ( ) 1 , 0 ( r c) R t t r ∈ + − = ), 0 , 2 ( ) 3 , 2 ( r Ex 2. Convert the symmetric equation of a line L : 2 3 1 2 − = − + y x to the vector equation. B Normal Equation Let consider a line L that passes through the specific point ) , ( 0 0 0 y x P and has the direction vector ) , ( y x u u u = r . The vectors ) , ( ) , ( B A u u n x y = − = r or ) , ( ) , ( B A u u n x y = − = r are perpendicular to the vector u r and so they are perpendicular to the line L . These are called normal vectors to the line L . Let ) , ( y x P be a generic point on the line L . So: 0 ) ( 0 || 0 0 0 0 = ⋅ − = ⋅ ⇒ ⊥ ⇒ n r r n P P n P P u P P r r r r r r The normal equation of a line is given by: 0 ) ( 0 = ⋅ − n r r r r r C Cartesian Equation The normal equation can be written as: 0 0 0 0 0 0 0 0 0 0 ) , ( ) , ( ) , ( ) , ( 0 By Ax C where C By Ax By Ax By Ax B A y x B A y x n r n r − − = = + + = − − + = ⋅ − ⋅ = ⋅ − ⋅ r r r r The Cartesian equation of a line is given by: 0 = + + C By Ax where ) , ( B A n = r is a normal vector and the constant C depends on a specific point of the line.
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Transcript of h82_Cartesian_Equation_of_a_Line.pdf
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
