Chapter 13: Vectors in Three Dimensional SpaceCopyright 2007 © John Wiley & Sons, Inc. All rights reserved.
Poglavlje 13: Vektori u trodimenzionalnom prostoru
Odjeljak 13.1 Kartezijev koordinatni sustav u prostoru
Pravilo desne ruke
Koordinate vektora
Teoremi
Udaljenost toke od pravca
Ravnina kroz tri nekolinearne toke
Udaljenost toke od ravnine
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Pravokutni koordinatni sustav u prostoru
Pravilo desne ruke: ako savijeni prsti desne ruke pokazuju od pozitivnog dijela x-osi prema pozitivnom dijelu y-osi (najmanjim kutem), palac pokazuje pozitivan smjer z-osi.
Toka na x-osi sa x-koordinatom x0 ima koordinate (x0, 0, 0);
toka na y-osi sa y-koordinatom y0 ima koordinate (0, y0, 0);
toka na z-osi sa z-koordinatom z0 ima koordinate (0, 0, z0).
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Kartezijev koordinatni sustav u prostoru
Postoje tri koordinatne ravnine; xy-ravnina, xz-ravnina, yz-ravnina.
Toka P u trodimenzionalnom prostoru ima koordinate (x0, y0, z0) ako
(1) ravnina kroz P paralelna sa yz-ravninom sijee x-os u (x0, 0,0),
(2) ravnina kroz P paralelna sa xz-ravninom sijee y-os u (0, y0,0),
(3) ravnina kroz P paralelna sa xy-ravninom sijee z-os u (0, 0,z0).
Koordinate x0, y0, z0 se zovu pravokutne koordinate od P, ili eše u ast
Descartesu, Kartezijeve koordinate od P.
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Kartezijev koordinatni sustav u prostoru
Formula za udaljenost toaka
Udaljenost d izmeu toaka i moe se dobiti koristei dva puta Pitagorin teorem.
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Kartezijev koordinatni sustav u prostoru
Sfera radijusa (polumjera) r sa središtem u P0(a, b, c) je skup toaka P(x, y, z) za koje je d(P, P0) = r . Za takve je toke [d(P, P0)]2 = r2. Primjenivši formulu za udaljenost dobivamo
Ovo je jednadba sfere radijusa r sa središtem u P0(a, b, c). Jednadba
predstavlja sferu radijusa r sa središtem u ishodištu.
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Kartezijev koordinatni sustav u prostoru
Simetrija
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Kartezijev koordinatni sustav u prostoru
Pravac kroz dvije toke
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Kartezijev koordinatni sustav u prostoru
Duina
Polovište
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Vektori u trodimenzionalnom prostoru
Vektor a za nas e biti ureena trojka realnih brojeva:
a = (a1, a2, a3).
Brojeve a1, a2, a3 zovemo komponente ili koordinate vektora a.
Dva vektora su jednaki ako imaju iste koordinate;
(a1, a2, a3) = (b1, b2, b3) ako i samo ako a1 = b1, a2 = b2, a3 = b3.
Zbrajanje vektora
Mnoenje vektora skalarom
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Vektori u trodimenzionalnom prostoru
Geometrijska interpretacija vektora
Orijentirane duine i su na razliitim pozicijama, ali budui da imaju istu duljinu, smjer i orijentaciju, predstavljaju isti vektor: vektor a = (a1, a2, a3).
Nulvektor 0 = (0, 0, 0) moemo predstaviti orijentiranom duinom duljine 0. Nulvektor nema definiran smjer i orijentaciju.
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Vektori u trodimenzionalnom prostoru
Grafiki prikaz a + b
Za a = (a1, a2, a3) i b = (b1, b2, b3) definiramo
a + b = (a1 + b1, a2 + b2, a3 + b3).
Pravilo trokuta i pravilo paralelograma
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Vektori u trodimenzionalnom prostoru
Svojstva norme:
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Vektori u trodimenzionalnom prostoru
Višekratnik vektora
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Vektori u trodimenzionalnom prostoru
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Vektori u trodimenzionalnom prostoru
Paralelni vektori
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Vektori u trodimenzionalnom prostoru
Primjer paralelnih vektora
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Vektori u trodimenzionalnom prostoru
Jedinini vektori
Vektore duljine 1 zovemo jedininim vektorima. Za svaki nenul vektor a postoji jedinstveni jedinini vektor ua s istim smjerom i orijentacijom kao a. To je
Vektori
i = (1, 0, 0), j = (0, 1, 0), k = (0, 0, 1).
su jedinini vektori koji su posebno korisni u raunu jer se svaki vektor a moe na jednostavan nain prikazati kao jedinstvena linearna kombinacija ovih vektora:
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Skalarni produkt
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Skalarni produkt
Svojstva skalarnog produkta
Ako skalarno pomnoimo vektor sa samim sobom dobivamo kvadrat njegove norme:
Dokaz
Dokaz
(a1)(0) + (a2)(0) + (a3)(0) = 0, (0)(a1) + (0)(a2) + (0)(a3) = 0.
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Skalarni produkt
distributivnost prema zbrajanju).
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Skalarni produkt
Geometrijska interpretacija skalarnog produkta
Poinjemo od trokuta sa stranicama duljine a, b, c. Ako je θ jednak ½π, Pitagorin teorem nam daje c2 = a2 + b2. Teorem o kosinusu je generalizacija za proizvoljni kut θ,
c2 = a2 + b2 − 2ab cos θ.
Skica dokaza teorema: Iz slike je y2 + x2 = a2 i y = a cos θ. Zato je
c2 = a2 + b2 − 2ab cos θ
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Skalarni produkt
Duljine stranica su ||a||, ||b||, ||a − b||. Teorem o kosinusu daje
||a − b||² = ||a||2 + ||b||2 − 2 ||a|| ||b|| cos θ.
No iskoristimo li da je ||a||2 = a a i slino za ostale dobivamo:
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Skalarni produkt
Dva vektora su okomiti ako je kut meu njima pravi ili ako je jedan od vektora nulvektor. Stoga su dva vektora okomiti ako i samo ako im je skalarni produkt jednak nula. Zapisano simbolima:
Jedinini vektori i, j, k su meusobno okomiti:
i · j = i · k = j · k = 0.
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Ortogonalna projekcija i komponente
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i, j, k komponente jedininog vektora su kosinusi smjera.
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Cauchy-Schwarz-Bunjakovski nejednakost
Budui je
a · b = ||a|| ||b|| cos θ,
gdje je θ kut izmeu a i b, uzmemo li apsolutne vrijednosti i iskoristimo
| cos θ| ≤ 1, dobivamo nejednakost
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Vektorski produkt
smjer
orijentacija
duljina
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Vektorski produkt
Desna trojka vektora
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Vektorski produkt
Mnoenjem bilo kojeg od triju vektora pozitivnim skalarom trojka ostaje desna: ako je (a, b, c) desna trojka i α > 0, onda su (αa, b, c), (a, αb, c), i (a, b, αc) takoer desne trojke.
Mnoenjem bilo kojeg od triju vektora negativnim skalarom trojka postaje lijeva: ako je (a, b, c) desna trojka vektora i α < 0, onda (αa, b, c), (a, αb, c), i (a, b, αc) nisu desne trojke (tj. to su lijeve trojke).
Svojstva desnih trojki vektora
I. Budui je (a × b) · (a × b) = ||(a × b)||2 > 0, (a, b, a × b) je desna trojka.
II. Ako je (a, b, c) desna trojka, onda su i (c, a, b) i (b, c, a) desne trojke.
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Vektorski produkt
Skalare moemo izluiti:
Vrijede obje distributivnosti:
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Vektorski produkt
Mješoviti produkt ili umnoak
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Vektorski produkt
Komponente od a × b
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Vektorski produkt
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Pravci
Vektorska jednadba
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Pravci
l1 : r(t) = r0 + td, l2 : R(u) = R0 + uD
sijeku se ako i samo ako postoje brojevi t i u takvi da je
r(t) = R(u).
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Pravci
Udaljenost od toke do pravca
Neka je P0 toka na pravcu l i neka je d vektor smjera od l. Ako su P0 i Q kao na slici, lako vidimo da je
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Ravnine
Skalarna jednadba ravnine
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Ravnine
Budui da je
(13.6.1) se moe zapisati u obliku
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Ravnine
Presjek ravnina
Jedinine normale
Ako je N vektor normale (tj. okomit) na danu ravninu, tada su svi ostali vektori normale na danu ravninu paralelni s N i zato skalarni višekratnici od N
Posebno, postoje dva jedinina vektora normale:
i
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Ravnine
Ravnina odreena trima nekolinearnim tokama
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Ravnine
Udaljenost toke od ravnine
(
)