Question

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

• A. a = 0
• B. a < 0
• C. a > 0
• D. None of these

Answer 1

Answer: (D)

$\vec{a}\cdot \vec{b}>0$
$\Rightarrow -6+a{\left({\log }_{3}x\right)}^2+a{\log }_{3}x>0$
$\Rightarrow a{\left({\log }_{3}x\right)}^2+a\left({\log }_{3}x\right)-6>0\dots (1)$
Discriminant $D=a^2+24a$
The sign of $D$ depends on a, so the inequality (1) can have solution for $a<0$ as well as $a>0$