If $$\bar{a}=2 \hat{i}-\hat{j}+\hat{k}, \bar{b}=\hat{i}+2 \hat{j}-3 \hat{k}$$ and $$\bar{c}=3 \hat{i}+\lambda \hat{j}+5 \hat{k}$$ are coplanar, then $$\lambda$$ is the root of the equation

If $$\hat{a}$$ is a unit vector such that $$(\bar{x}-\hat{a}) \cdot(\bar{x}+\hat{a})=8$$, then $$|\bar{x}|=$$

Let $$\vec{v}=2 \hat{i}+2 \hat{j}-\hat{k}$$ and $$\bar{w}=\hat{i}+3 \hat{k}$$. If $$\bar{u}$$ is a unit vector, then the maximum value of the scalar triple product $$[\bar{u} \bar{v} \bar{w}]$$ is

If $$\overrightarrow{\mathrm{a}}=\hat{\mathrm{i}}+2 \hat{\mathrm{j}}+3 \hat{\mathrm{k}}, \overline{\mathrm{b}}=-\hat{\mathrm{i}}+2 \hat{\mathrm{j}}+\hat{\mathrm{k}}, \overline{\mathrm{c}}=3 \hat{\mathrm{i}}+\hat{\mathrm{j}}$$ and $$\overline{\mathrm{a}}+\lambda \overline{\mathrm{b}}$$ is perpendicular to $$\overline{\mathrm{c}}$$, then $$\lambda=$$