MathematicsLet Sbe the set of all a ∈R for which the angle betwen the vectors u⇀=alogebi^-6j^+3k^ andv→=logebi^+2j^+2alogebk^ ,b>1is acute. Then S is equal to

Let Sbe the set of all a $\in$ R for which the angle betwen the vectors $\overset{⇀}{u} = a (\log_{e} b) \hat{i} - 6 \hat{j} + 3 \hat{k}$ and
$\vec{v} = (\log_{e} b) \hat{i} + 2 \hat{j} + 2 a (\log_{e} b) \hat{k}, (b > 1)$ is acute. Then S is equal to

A
$(\infty, - \frac{4}{3})$
B
$ϕ$
C
$(- \frac{4}{3}, 0)$
D
$(\frac{12}{7}, \infty)$

Solution:

Angle between the vectors $\vec{u}, \vec{v}$ is acute

$\Rightarrow \vec{u} . \vec{v} > 0 \Rightarrow a {(\log_{e} b)}^{2} - 12 + 6 a (\log_{e} b) > 0 \forall b > 1 l e t \log_{e} b = t \Rightarrow t > 0 a s b > 1 y = a t^{2} + 6 a t - 12 & y > 0, \forall t > 0 a > 0, D < 0 \Rightarrow 36 a^{2} + 48 a < 0 w h i c h i s n o t p o s s i b l e F o r a < 0, D < 0 \Rightarrow 36 a^{2} + 48 a < 0 \Rightarrow 12 a (3 a + 4) < 0 \frac{- 4}{3} < a < 0 \Rightarrow y < 0 ∴ a \in ϕ$

Let Sbe the set of all a $\in$ R for which the angle betwen the vectors $\overset{⇀}{u} = a (\log_{e} b) \hat{i} - 6 \hat{j} + 3 \hat{k}$ and
$\vec{v} = (\log_{e} b) \hat{i} + 2 \hat{j} + 2 a (\log_{e} b) \hat{k}, (b > 1)$ is acute. Then S is equal to

Solution:

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