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Q.
If the vectors $\vec{ a }=\left(2, \log _{3} x , a \right)$ and
$\vec{ b }=\left(-3, a \log _{3} x , \log _{3} x \right)$ are inclined at an acute angle then
Vector Algebra
Solution:
$\vec{a} \cdot \vec{b}>\,0$
$ \Rightarrow -6+a\left(\log _{3} x\right)^{2}+a \log _{3} x>\,0$
$\Rightarrow a\left(\log _{3} x\right)^{2}+a\left(\log _{3} x\right)-6>\,0\,\,\,\,\dots(1)$
Discriminant $D = a ^{2}+24 a$
The sign of $D$ depends on a, so the inequality (1) can have solution for $a<\,0$ as well as $a>\,0$