Pismeni zadaci

Zadatak 2020

Test: MPsIIIVAII_350

 

N Dati su vektori \(\overrightarrow{a}=(m-1,1,m+1),\overrightarrow{b}=(2,3,1)\) i \(\overrightarrow{c}=(1,3,2)\). Odredi \(m\) tako da vektor \(\overrightarrow{a}\times (\overrightarrow{b}\times \overrightarrow{c})\) bude normalan na vektor \(\overrightarrow{b}-\overrightarrow{c}\) i za tako dobijenu vrednost parametra \(m\) odredi zapreminu paralelopipeda konstruisanog nad vektorima \(\overrightarrow{a},\overrightarrow{b}\) i \(\overrightarrow{c}\).

Zadatak 2019

Test: MPsIIIVAII_350

 

S Odredi kosinus ugla između vektora \(\overrightarrow{a}\) i \(\overrightarrow{b}\) ako su vektori \(2\overrightarrow{a}+\overrightarrow{b}\) i \(\overrightarrow{a}-2\overrightarrow{b}\) a takođe i vektori \(2\overrightarrow{a}-\overrightarrow{b}\) i \(\overrightarrow{a}+\overrightarrow{b}\) uzajamno ortogonalni.

Zadatak 1941

Test: MPsIIIVAII_334

 

S Dati su vektori \(\overrightarrow{a}=\overrightarrow{m}+3\overrightarrow{n},\overrightarrow{b}=7\overrightarrow{m}-5\overrightarrow{n},\overrightarrow{c}=\overrightarrow{m}-4\overrightarrow{n}\) i \(\overrightarrow{d}=7\overrightarrow{m}-2\overrightarrow{n}\). Ako je \(\overrightarrow{m}\perp \overrightarrow{n}\) i \(\overrightarrow{c}\perp \overrightarrow{d}\):

a) dokaži da je \(\left | \overrightarrow{m} \right |=\left | \overrightarrow{n} \right |\);

b) odredi ugao između vektora \(\overrightarrow{m}\) i \(\overrightarrow{n}\).

Zadatak 1942

Test: MPsIIIVAII_334

 

S Dati su vektori \(\overrightarrow{a}=(1,0,4),\overrightarrow{b}=(1,-1,-2)\) i \(\overrightarrow{c}=(0,1,-3)\). Izračunaj:

a) $$(\overrightarrow{a}+2\overrightarrow{b}+3\overrightarrow{c})\cdot ((\overrightarrow{a}+\overrightarrow{b})\times (\overrightarrow{a}+\overrightarrow{b}+\overrightarrow{c}))$$

b) $$(\overrightarrow{a}+\overrightarrow{b})\cdot ((\overrightarrow{a}-\overrightarrow{c})\times \overrightarrow{b})$$

Zadatak 1864

Test: MPsIIIVAII_318

 

N Dokaži da za sve vektore \(\vec{a},\vec{b},\vec{c}\) važi: $$\left [ \left ( \vec{a}+\vec{b} \right )\times \left ( \vec{b}+\vec{c} \right ) \right ]\cdot (\vec{c}+\vec{d})=2\vec{c}\cdot (\vec{b}\times \vec{d})$$

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