Question

Sabendo que o ângulo entre os vetores = (2, 1, −1) e = (1, −1, m + 2) é pi/3

, determine m

Answer 1

Olá, Larissa.

 

$<var>u= (2, 1, -1), v= (1, 1, m + 2) \\\\ \cos \frac{\pi}{3} = \frac{u \cdot v} {|u| \cdot |v|} \Rightarrow \cos 60\º=\frac{2\cdot1+1\cdot1-1\cdot(m+2)}{\sqrt{2^2+1^2+(-1)^2}\cdot \sqrt{1^2+1^2+(m+2)^2}} \Rightarrow \\\\ \frac12=\frac{3-m-2}{\sqrt6 \cdot \sqrt{2+m^2+4m+4}} \Rightarrow \frac14=\frac{1-2m+m^2}{6(m^2+4m+6)} \Rightarrow \\\\ 4-8m+4m^2=6m^2+24m+36 \Rightarrow 2m^2+32m+32=0 \Rightarrow\\\\ m^2+16m+16=0 \Rightarrow m=\frac{-16\pm\sqrt{192}}{2}=\frac{-16\pm8\sqrt{3}}{2}\\\\ </var>$

 

$\therefore\boxed{m=-8\pm4\sqrt3}$