Question

Sabe-se que o ângulo entre dois vetores u= (2,1,-1) e v= (1, -1, m + 2) é 60º. Qual é o valor de m ?

Answer 1

Olá, Jardeson.

 

$<var>u= (2,1,-1), v= (1, -1, m + 2) \\\\ u\cdot v=2\cdot1-1\cdot1-1\cdot(m+2)=2-1-m-2=-m-1 \\\\ |u|=\sqrt{2^2+1^2+1^2}=\sqrt6 \\\\ |v|=\sqrt{1^2+1^2+(m+2)^2}=\sqrt{(m+2)^2+2} \\\\ \cos 60\º= \frac{u \cdot v}{|u| \cdot |v|} \Rightarrow \frac12=\frac{-(m+1)}{\sqrt6(\sqrt{(m+2)^2+2})} \Rightarrow \\\\ \frac14=\frac{(m+1)^2}{6((m+2)^2+ 2)} \Rightarrow \frac32=\frac{m^2+2m+1}{m^2+4m+6} \Rightarrow \\\\ 3m^2+12m+18=2m^2+4m+2 \Rightarrow m^2+8m+16=0 \Rightarrow \\\\ (m+4)^2=0\\\\ \therefore \boxed{m=-4} </var>$