Question

Let $S$ be the set of all $a \in R$ for which the angle between the vectors $\overrightarrow{ u }= a \left(\log _e b \right) \hat{ i }-6 \hat{ j }+3 \hat{ k }$ and $\vec{v}=\left(\log _e b\right) \hat{i}+2 \hat{j}+2 a\left(\log _e b\right) \hat{k},(b>1)$ is acute. Then $S$ is equal to

• A. $\left(-\infty,-\frac{4}{3}\right)$
• B. $\Phi$
• C. $\left(-\frac{4}{3}, 0\right)$
• D. $\left(\frac{12}{7}, \infty\right)$

Answer 1

Answer: (B)

For angle to be acute
$ \overrightarrow{ u } \cdot \overrightarrow{ v }>0 $
$ \Rightarrow a \left(\log _{ e } b \right)^2-12+6 a \left(\log _{ e } b \right)>0$
$ \forall b >1 $
$ \text { let } \log _{ e } b = t \Rightarrow t >0 \text { as } b >1 $
$ y = at ^2+6 at -12 \& y >0, \forall t >0 $
$ \Rightarrow a \in \phi$