If the vectors $\overline{\mathrm{a}}=\hat{\mathrm{i}}-\hat{\mathrm{j}}+2 \hat{\mathrm{k}}, \overline{\mathrm{b}}=2 \hat{\mathrm{i}}+4 \hat{\mathrm{j}}+\hat{\mathrm{k}}$ and $\overline{\mathrm{c}}=\mathrm{pi}+\hat{\mathrm{j}}+\mathrm{q} \hat{\mathrm{k}}$ are mutually orthogonal, then $(p, q)$ is equal to

If $\bar{u}, \bar{v}$ and $\bar{w}$ are three non-coplanar vectors, then $(\bar{u}+\bar{v}-\bar{w}) \cdot[(\bar{u}-\bar{v}) \times(\bar{v}-\bar{w})]$ is equal to

Let $\overline{\mathrm{u}}=\hat{\mathrm{i}}+\hat{\mathrm{j}}, \overline{\mathrm{v}}=\hat{\mathrm{i}}-\hat{\mathrm{j}}$ and $\overline{\mathrm{w}}=\hat{\mathrm{i}}+2 \hat{\mathrm{j}}+3 \hat{\mathrm{k}}$. If $\hat{\mathrm{n}}$ is a unit vector such that $\overline{\mathbf{u}} \cdot \hat{\mathrm{n}}=0$ and $\overline{\mathrm{v}} \cdot \hat{\mathrm{n}}=0$, then $|\overline{\mathrm{w}} \cdot \hat{\mathrm{n}}|$ is equal to

Let $\bar{a}, \bar{b}$ and $\bar{c}$ be three unit vectors such that $\overline{\mathrm{a}} \times(\overline{\mathrm{b}} \times \overline{\mathrm{c}})=\frac{\sqrt{3}}{2}(\overline{\mathrm{~b}}+\overline{\mathrm{c}})$. If $\bar{b}$ is not parallel to $\bar{c}$, then the angle between $\bar{a}$ and $\bar{b}$ is