MATH1131 Lecture Notes - Lecture 4: Cartesian Coordinate System, Enol, Tibet
3 Jun 2018
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Department
Course
Professor
5. E :2r * 1.,2 :2
1.5 Planes
Pla,iies are the 2 dimensiorral analogues of linr.ls.
Before r,ve begin studying pla,nes.'wr,: rurecl to k:nm soure
new termirxrlogy.
Definition 1.34. Let v,v1,v2 € IRn. We say v is
o linear combinatiort, orin the span, of v1,v2, if
there etists a 11,.\2 €R such that
v: }rvr*),zvz.
in the span of
uector of some point on the plane, o.ndv1
the d,irect'ions of the plane. The two d'i
not be parallel.
Definition
form in lR" r,s any erpress'ion af the form
x:a*)1v1 *)2v
-
.for some .\1,,\2 € R uthere a € iRr? is coordinate
,411u t* d#rnn{ lwthont
L,36. A plane in irarametric vector
(:)-
,,'(j) (;')'
\
z € ]R" are\i
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Ifthe two rlircctions arc palalkl. lhcn th<r lttation rbfirr:s
a linr:, not a planc. Let's sec what thi kroks lil<c with
whcn
v e span(v1,v2).
Example 1.37. Describr: the geornetry of the follow-
ing oh.jr'r'ts:
i,ons must
, ,\1,,\2 € IR.
1. x:
Alo { (+)
( ;,).^, ( ,:)
-i) - nt gnllrl
(i)
nt , ftn
*lr
((
[j) -'
M[ [, [i)
Example 1.35.
' ,' ( i)
(+),
[rnU (flS
a lineal comltination of
ooor (i) = ,,()J . n,
ftr rsmt v1 , 7t. g,p 1-
,, (',l,l'') , +h',' *,,
whrn lr= -z an4 il, =3 , &q und nr+3Ar= 3-6=*
Yo, r! (i)=,(i) _.ti)
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I
t:L-
to
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