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If the vectors $a = (c \log_{2} x) i - 6 j + 3 k$ and
$b = (\log_{2} x) i + 2 j + (2 c \log_{2} x) k$ make an obtuse angle for any $x \in (0, \infty),$ then $c$ belongs to

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$(0, \frac{4}{3})$

$(- \infty, - \frac{4}{3})$

$(- \frac{4}{3}, 0)$

$(- \frac{4}{3}, \infty)$

answer is C.

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Detailed Solution

$a . b < 0 \Rightarrow {(\log_{2}^{x})}^{2} - 12 + 6 c \log_{2}^{x} < 0 ∆ < 0 \Rightarrow 36 c^{2} - 4 c (- 12) < 0 3 c^{2} + 4 c < 0 c (3 c + 4) < 0 c \in (\frac{- 4}{3}, 0)$ Question Image

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