JEE Mains · Maths · STD 12 - 10. vector algebra
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 & Solution
Correct Answer
(B) \(\phi\)
Step-by-step Solution
Detailed explanation
Sol. For angle to be acute \(\overrightarrow{ u } \cdot \overrightarrow{ v }>0\) \(a \left(\log _{ e } b \right)^{2}-12+6 a \left(\log _{ e } b \right)>0\) \(\forall b >1\) let \(\log _{ e } b = t \Rightarrow t >0\) as \(b >1\) \(y =a at ^{2}+6 a at -12 \& y >0, \forall t >0\)4…
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