Question

Sa se arate ca vectorii v=5i-4j si u=2i+3j formeaza un unghi obtuz

Answer 1

[tex]Trebuie\ sa\ calculam\ cosinusul\ unghiului\ dintre\ cei\ doi\ vectori\\ iar\ daca\ acesta\ este\ mai\ mic\ ca\ 0\ atunci\ unghiul\ este\ obtuz.\\ Pentru\ aceasta\ vom\ folosi\ formula:\\ \\ \boxed{\cos(\vec{u};\vec{v})=\frac{a_1\cdot a_2+b_1\cdot b_2}{\sqrt{a_1^2+b_1^2}\cdot\sqrt{a_2^2+b_2^2}}}\\ Inlocuim:\\ \cos(\vec{u};\vec{v})=\frac{5\cdot2-4\cdot3}{\sqrt{25+16}\cdot\sqrt{4+9}}\\ \\ \cos(\vec{u};\vec{v})=\frac{10-12}{\sqrt{41}\cdot\sqrt{13}}\\ \\ \cos(\vec{u};\vec{v})=-\frac{2}{\sqrt{533}}\\ [/tex]
[tex]Dar\ se\ vede\ ca:\\ -\frac{2}{\sqrt{533}}\ \textless \ 0\Rightarrow\cos(\vec{u};\vec{v})\ \textless \ 0.\\ Asadar\ unghiul\ este\ obtuz.[/tex]