Let S be the set of all a $\in R$ for which the angle between the vectors $\vec{u}=a\left(\log _{e} b\right) \hat{i}-6 \hat{j}+3 \hat{k}$ 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 :
Solution
<p>$\overrightarrow u = a({\log _e}b)\widehat i - 6\widehat j + 3\widehat k$</p>
<p>$$\overrightarrow v = ({\log _e}b)\widehat i + 2\widehat j + 2a({\log _e}b)\widehat k$$</p>
<p>For acute angle $\overrightarrow u \,.\,\overrightarrow v > 0$</p>
<p>$\Rightarrow a{({\log _e}b)^2} - 12 + 6a({\log _e}b) > 0$</p>
<p>$\because$ $b > 1$</p>
<p>Let ${\log _e}b = t \Rightarrow t > 0$ as $b > 1$</p>
<p>$a{t^2} + 6at - 12 > 0\,\,\,\,\,\,\,\forall t > 0$</p>
<p>$\Rightarrow a \in \phi$</p>
About this question
Subject: Mathematics · Chapter: Vector Algebra · Topic: Types of Vectors
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