If the angle between the vectors \(\bar{a}=2 \lambda^2 \hat{i}+4 \lambda \hat{j}+\hat{k}\) and \(\overline{\mathrm{b}}=7 \hat{\mathrm{i}}-2 \hat{\mathrm{j}}+\lambda \hat{\mathrm{k}}\) is obtuse, then \(\lambda \in\)

We have \(\bar{a}=2 \lambda^2 \hat{i}+4 \lambda \hat{j}+\hat{k}\) and \(\bar{b}=7 \hat{i}-2 \hat{j}+\lambda \hat{k}\) \(\bar{a} \cdot \bar{b}=|\bar{a}| \cdot|\bar{b}| \cdot \cos \theta\), where \(\theta\) is angle between \(\bar{a}\) and \(\bar{b}\).
\(\therefore \quad \cos \theta=\frac{\overline{\mathrm{a}} \cdot \overline{\mathrm{b}}}{|\overline{\mathrm{a}}| \cdot|\overline{\mathrm{b}}|} < 0\), as \(\theta\) is an obtuse angle.
\(\therefore \quad|\overline{\mathrm{a}}| \cdot|\overline{\mathrm{b}}| < 0\)
\(\therefore \quad\left(2 \lambda^2\right)(7)+(4 \lambda)(-2)+(1)(\lambda) < 0\)
\(\therefore \quad 14 \lambda^2-7 \lambda < 0 \Rightarrow 7 \lambda(2 \lambda-1) < 0 \Rightarrow \lambda(2 \lambda-1) < 0\)
\(\therefore \quad 0 < \lambda < \frac{1}{2}\) i.e. \(\lambda \in\left(0, \frac{1}{2}\right)\)