IE Ft29y=mxtn n # q=pyq Lines -1=5 +2 : egg line LY - - PDF document

SLIDE 1

• EegeometyaadlinearaCge8#

1/3

Vectors

in the place

:

points in the plane

:

• 44

a

" a

i¥÷÷i÷i÷÷÷:÷÷÷

: :÷÷÷÷÷÷÷.

• a generalization of a point

,

'Et

Y

y

a-

• up )

I

• l's)

we have:S

• Eta

E- l'd

ftp.HH-EI.is)

{ =L})

Or in other words : a-

= I -I

=

n

µFt29y=mxtn

Lines

:

#

egg

• q=pyq
• 1=5×+2

IE

z -

p

y

line LY

eR4y=Ixtz}

i-TH

§=(E) But Houtorepreseat vertical lines?

E- E)

E-if

HEH'D

Another wayto describe this line :

1=44

• steps -51 SERJ =L Fts
• (off)

( SERJ

• IE

affine combination

direction vector

=

=L Sipe Sig (

syszEIRausitsz-BLiheseguey-tp-g-LU-slpts.ci/st5o,D

}

• 5-0 : Fto
• g- =D

⇐ 0.4 :

. . .

" liheerihtrpoktionfetveeg

5- I

:

• .p-ts.gr
• _ of

Baudot

4

Three points;pTgYT : Trieste

• Lsiptsz
• g- TST

/ sysz, Szeto,D

and S , -152-(53--1)

pain

. .

mass

.

III:

÷

:*,

npointsp,h'

• CHkpyypidqffgsipilsittoidau.ES

it}

Alternative characterization

. Need to prove equihua to our other definition .

SLIDE 2

Orientation testIhalfplaukst ;

2/3

• e.EE#::e.EiLeftthm(

positive

)

Right turn ( negating

zero

relies

• n leffsideofl

rliesoh right side off

usethesignoftheonertedareaofspqr.LT

, #

E-it

"

TE

Yam:p:3:£m

?

b-

i'

signatory

's

:{¥:{÷÷I:&

.me

.

ssajoqsyeaseiscouud.am

signed area of parallelogram

• det (I}

= ax.by

• ay
• K ⑧

qx

• ok

9

⇒ signed

area of spgrt-tzdetfagYJJ-tzdetfgy.ie

, I

' '

• I
• Ugx
• e) fry
• agg)
• lay
• py )
• (4-9×1)

⑧

• ztfgkry-qgrx-fpxry-pyrxp-pxgg-qxpyJ-aa.li#..::.l--tl'

Might

\

'

÷i÷ti÷÷÷

:* .

÷a÷:¥÷¥iH÷

Area of

• fxtsxkagtbg)
• I

by

• b

y

#

• xay
• I

ay

• sx⑤ag
• though

= axby

• bxay

SLIDE 3

Scakrprodhct

: f. of

= ( Ppt )

. ( 9g! )

=

pig

, t prop

3/3

Length of

a

vector :

l l F H = pv.fm

Geometric

interpretation f. of = HOTH

. It pull .

Cos a

¥÷:>

9

• projection ,

a,¥w%

• f p onto 9
• pposite

side cos ← adjacent

⇒

F

• 5=115/1

. Npg . cosy

a

*potehhse

⇒a=cosj¥

Works

also in Rd

• Another characterization of a line l ;

e-

• e.

peril t.EE?s--tF4R4 "÷÷÷÷f¥yI

• constant

set of all

vectors whose projection

• nto g- has fixed length

it

, e

⇒ Same definition works in Rd to

characterize hyperplane

using their normal vector

• 9

.