Let S be the set of all a $\in$ R for which the angel between the vectors $\overrightarrow{u}=a\left(lo{g}_{e}b\right)\widehat{i}-6\widehat{j}+3\widehat{k}$ and $\overrightarrow{v}=\left(lo{g}_{e}b\right)\widehat{i}+2\widehat{j}+2a\left(lo{g}_{e}b\right)\widehat{k},\left(b>1\right)$ is acute. Then S is equal to:

<p>For angle to the acute&nbsp;<math><mrow><mover accent="true"><mi>U</mi><mo>→</mo></mover><mo>.</mo><mover accent="true"><mi>V</mi><mo>→</mo></mover><mo>&gt;</mo><mn>0</mn></mrow></math></p><p><math><mrow><mo>⇒</mo><mi>a</mi><msup><mrow><mrow><mo> (</mo><mrow><mi>l</mi><mi>o</mi><msub><mrow><mi>g</mi></mrow><mrow><mi>e</mi></mrow></msub><mi>b</mi></mrow><mo>)</mo></mrow></mrow><mrow><mn>2</mn></mrow></msup><mo>−</mo><mn>1</mn><mn>2</mn><mo>+</mo><mn>6</mn><mi>a</mi><mrow><mo> (</mo><mrow><mi>l</mi><mi>o</mi><msub><mrow><mi>g</mi></mrow><mrow><mi>e</mi></mrow></msub><mi>b</mi></mrow><mo>)</mo></mrow><mo>&gt;</mo><mn>0</mn></mrow></math></p><p><math><mrow><mo>∀</mo><mi>b</mi><mo>&gt;</mo><mn>1</mn></mrow></math></p><p>Let log_eb = t &gt; 0 as b &gt; 1</p><p>Y = at² + 6at – 12 &amp; y &gt; 0 <math><mrow><mo>∀</mo></mrow></math>&nbsp;t &gt; 0</p><p><math><mrow><mo>⇒</mo><mi>a</mi><mo>∈</mo><mi>? </mi></mrow></math></p>

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