Licenca
To delo je na voljo pod pogoji slovenske licence Creative Commons 2.5:

priznanje avtorstva - nekomercialno - deljenje pod enakimi pogoji.

Celotna licenca je na voljo na spletu na naslovu http://creativecommons.org/licenses/by-nc-sa/2.5/si/. V skladu s to licenco je dovoljeno vsakemu uporabniku delo razmnoževati, distribuirati, javno priobčevati, dajati v najem in tudi predelovati, vendar samo v nekomercialne namene in ob pogoju, da navede avtorja oziroma avtorje in izdajatelja tega dela. Če uporabnik delo predela, kar pomeni, da ga spremeni, preoblikuje, prevede ali uporabi to delo v svojem delu, lahko predelavo dela ponudi na voljo le pod pogoji, ki so enaki pogojem iz te licence oziroma pod enako licenco.

Za kolinearna vektorja $\overset{\rightharpoonup}{a},\overset{\rightharpoonup}{b}$ velja:

Vektorja sta kolinearna natanko takrat, kadar je njun vektorski produkt enak $\overset{\rightharpoonup}{0}$.

Izračunaj vektorske produkte $\overset{\rightharpoonup}{i}\times\overset{\rightharpoonup}{i},\overset{\rightharpoonup}{j}\times\overset{\rightharpoonup}{j},\overset{\rightharpoonup}{k}\times\overset{\rightharpoonup}{k},\overset{\rightharpoonup}{i}\times\overset{\rightharpoonup}{j},\overset{\rightharpoonup}{j}\times\overset{\rightharpoonup}{k},\overset{\rightharpoonup}{k}\times\overset{\rightharpoonup}{i}$.

Za bazne vektorje $\overset{\rightharpoonup}{i},\overset{\rightharpoonup}{j},\overset{\rightharpoonup}{k}$ velja:$$\overset{\rightharpoonup}{i}\times\overset{\rightharpoonup}{i}=\overset{\rightharpoonup}{j}\times\overset{\rightharpoonup}{j}=\overset{\rightharpoonup}{k}\times\overset{\rightharpoonup}{k}=\overset{\rightharpoonup}{0}$$ $$\overset{\rightharpoonup}{i}\times\overset{\rightharpoonup}{j}=\overset{\rightharpoonup}{k},\overset{\rightharpoonup}{j}\times\overset{\rightharpoonup}{k}=\overset{\rightharpoonup}{i},\overset{\rightharpoonup}{k}\times\overset{\rightharpoonup}{i}=\overset{\rightharpoonup}{j}$$

V nadaljevanju si bomo ogledali, kake lastnosti ima vektorski produkt.

Lastnosti vektorskega produkta

Premisli, v katero smer kažeta vektorja $\overset{\rightharpoonup}{a}\times\overset{\rightharpoonup}{b}$ in $\overset{\rightharpoonup}{b}\times\overset{\rightharpoonup}{a}$.

Vektorja $\overset{\rightharpoonup}{a}\times\overset{\rightharpoonup}{b}$ in $\overset{\rightharpoonup}{b}\times\overset{\rightharpoonup}{a}$ sta:

Lastnosti vektorskega produkta

  • $\overset{\rightharpoonup}{b}\times\overset{\rightharpoonup}{a}=-\overset{\rightharpoonup}{a}\times\overset{\rightharpoonup}{b}$ antikomutativnost
  • $(m\overset{\rightharpoonup}{a})\times\overset{\rightharpoonup}{b}=m(\overset{\rightharpoonup}{a}\times\overset{\rightharpoonup}{b})=\overset{\rightharpoonup}{a}\times(m\overset{\rightharpoonup}{b})$ homogenost
  • $\overset{\rightharpoonup}{a}\times(\overset{\rightharpoonup}{b}+\overset{\rightharpoonup}{c})=\overset{\rightharpoonup}{a}\times\overset{\rightharpoonup}{b}+\overset{\rightharpoonup}{a}\times\overset{\rightharpoonup}{c}$ distributivnost
  • $(\overset{\rightharpoonup}{a}+\overset{\rightharpoonup}{b})\times\overset{\rightharpoonup}{c}=\overset{\rightharpoonup}{a}\times\overset{\rightharpoonup}{c}+\overset{\rightharpoonup}{b}\times\overset{\rightharpoonup}{c}$ distributivnost

Zgled

Izračunaj $(2\overset{\rightharpoonup}{a}-3\overset{\rightharpoonup}{b})\times (4\overset{\rightharpoonup}{a}+5\overset{\rightharpoonup}{b})$.

<NAZAJ
>NAPREJ326/703