If $\bar{a}=(2 \hat{i}+2 \hat{j}+3 \hat{k}), \vec{b}=(-\hat{i}+2 \hat{j}+\hat{k}) \quad$ and $\bar{c}=(3 \hat{i}+\hat{j})$ such that $(\bar{a}+\lambda \bar{b})$ is perpendicular to $\bar{c}$, then the value of $\lambda$ is

If $x_0$ is the point of local minima of $f(x)=\overline{\mathrm{a}} \cdot(\overline{\mathrm{b}} \times \overline{\mathrm{c}})$ where $\overline{\mathrm{a}}=x \hat{\mathrm{i}}-2 \hat{\mathrm{j}}+3 \hat{\mathrm{k}}$, $\overline{\mathrm{b}}=-2 \hat{\mathrm{i}}+x \hat{\mathrm{j}}-\hat{\mathrm{k}}, \overline{\mathrm{c}}=7 \hat{\mathrm{i}}-2 \hat{\mathrm{j}}+x \hat{\mathrm{k}}$, then value of $\overline{\mathrm{a}} \cdot \overline{\mathrm{b}}$ at $x=x_0$ is

$\hat{a}, \hat{b}$, and $\hat{c}$ are three unit vectors such that $\hat{a} \times(\hat{b} \times \hat{c})=\frac{\sqrt{3}}{2}(\hat{b}+\hat{c})$. If $\dot{b}$ is not parallel to $\hat{c}$, then the angle between $\hat{a}$ and $\hat{b}$ is

For all real $x$, the vectors $C x \hat{i}-6 \hat{j}-3 \hat{k}$ and $x \hat{\mathrm{i}}+2 \hat{\mathrm{j}}+2 \mathrm{C} x \hat{\mathrm{k}}$ make an obtuse angle with each other, then the value of C can be in