Question

O vetor = ( 2,1,-1) forma um ângulo de 60º com o vetor determinado pelos pontos X(3,1,-2) e Y(4,0,a). Qual o valor de a?

Answer 1

$u = (2,1,-1) \\ \\ v=(1,-1,a+2) \\ \\ \\ \cos \theta = \displaystyle \frac{u \cdot v}{||u|| \cdot ||v||} \\ \\ \\ \frac{1}{2}=\frac{(2,1,-1) \cdot (1,-1,a+2)}{ \sqrt{2^{2}+1^{2}+(-1)^{2}} \cdot \sqrt{1^{2}+(-1)^{2}+(a+2)^{2}} } \\ \\ \\ \frac{1}{2}= \frac{-1-a}{ \sqrt{6} \cdot \sqrt{a^{2}+4a+6} } \\ \\ \\ \frac{1}{2}= \frac{-1-a}{ \sqrt{6a^{2}+24a+36} } \\ \\ \\ ( \frac{1}{2} )^{2} = (\frac{-1-a}{\sqrt{6a^{2}+24a+36}})^{2} \\ \\ \\ \frac{1}{4}= \frac{a^{2}+2a+1}{6a^{2}+24a+36}}$

$6a^{2}+24a+36=4a^{2}+8a+4 \\ \\ \\ 6a^{2}+24a+36-4a^{2}-8a-4=0 \\ \\ \\ 2a^{2}+16a+32=0 \\ \\ \\ \boxed{\boxed{a=-4}}$