Find the values of $c$ for which the vectors $\vec a=(c\log_2x) \hat i-6\hat j+3\hat k$ and $\vec b=(\log_2x)\hat i-2\hat j+(2c\log_2x)\hat k$ make an obtuse angle for any $x {\epsilon} (0,\infty)$
I found out the dot product, and found $cos\theta<0$, whereby I got a quadratic inequality. Since, the inequality was less than zero, I used the discriminant as not having any solution to find the value of $c \epsilon (0,\frac{4}{3})$. However, the answer did not match, and I am a bit confused. Please help.
we have $$\cos(\phi)=\frac{\vec{a}\cdot \vec{b}}{|\vec{a}||\vec{b}|}$$ so we get $$\vec{a}\cdot \vec{b}=c(\log_2 x)^2+12+6c\log_2 x$$ can you finish? the discriminat is $$9-\frac{12}{c}$$ it must be $$9-\frac{12}{c}<0$$ and we get $$0<c<\frac{4}{3}$$ as you stated