If the vectors $\vec a = (c\, \log_2 x)\hat i -6\hat j + 3\hat k$ and $\vec b=(\log_2 x)\hat i+2\hat j+(2c\, \log_2 x)\hat k$ make an obtuse angle for any $x ∈ (0, ∞)$, then c belongs to

(-4/3, 0)

For the vectors $\vec a$ and $\vec b$ to be inclined at an obtuse angle, we must have

$\vec a. \vec b <0$ for all $x ∈ (0,∞)$

$⇒c (\log_2 x)^2-12+6 c (\log_2 x) <0$ for all $x ∈ (0,∞)$

$⇒cy^2+6 cy-12 <0$ for all $y ∈ R$, where $y = \log_2 x$

$⇒c<0$ and $36 c^2 + 48 c <0$

$⇒c<0$ and $c(3c +4) <0$

$⇒c<0$ and $-\frac{4}{3}<c <0$

$⇒c∈ (-4/3, 0)$