Let S be the set of all a ∈ R for which the angle between the...

Given,

\overset{\to}{u} = a\left(\log_{e}b\right)\overset{^}{i} - 6\overset{^}{j} + 3\overset{^}{k} and \overset{\to}{v} = \left(\log_{e}b\right)\overset{^}{i} + 2\overset{^}{j} + 2a\left(\log_{e}b\right)\overset{^}{k}

For angle to be acute $\overset{\to}{u} \cdot \overset{\to}{v} > 0$

⇒ \left(a\left(\log_{e}b\right)\overset{^}{i} - 6\overset{^}{j} + 3\overset{^}{k}\right) . \left(\left(\log_{e}b\right)\overset{^}{i} + 2\overset{^}{j} + 2a\left(\log_{e}b\right)\overset{^}{k}\right) > 0

$⇒ a{\left(\log_{e}b\right)}^{2} - 12 + 6a\left(\log_{e}b\right) > 0$ $\forall b > 1$

Let $y = a{\left(\log_{e}b\right)}^{2} - 12 + 6a\left(\log_{e}b\right)$

Now let $\log_{e}b = t ⇒ t > 0$ as $b > 1$

Now $y = a{t}^{2} + 6at - 12 > 0, \forall t > 0$

Since $y > 0$, so $a > 0 \text{ and } D < 0$

$⇒ D = 36{a}^{2} + 48a < 0$

$⇒ a \in$\left(\dfrac{- 4}{3}, 0\right)$but$a > 0$,</p><p>So,$a \in ϕ