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If angle between the vectors $\bar{a}=2 \lambda 2 \hat{i}+4 \lambda \hat{j}+\widehat{k}$ and $\bar{b}=7 \hat{i}-2 \hat{j}+\lambda \widehat{k} i$ obtuse, then the values of $\lambda$ lie in
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The correct answer is:
$\left(0, \frac{1}{2}\right)$
Angle between $\vec{a}$ and $\vec{b}$ is obtuse
$\begin{aligned} & \Rightarrow(2 \lambda 2 \hat{i}+4 \lambda \hat{j}+\widehat{k}) \cdot(7 \hat{i}-2 \hat{j}+\lambda \widehat{k})<0 \\ & \Rightarrow 14 \lambda^2-8 \lambda+\lambda<0 \\ & \Rightarrow 7 \lambda(2 \lambda-1)<0 \\ & \Rightarrow \lambda \in\left(0, \frac{1}{2}\right)\end{aligned}$
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