Let $\bar{a}$ and $\bar{b}$ be two vectors such that $|\overline{\mathrm{a}}|=1,|\overline{\mathrm{~b}}|=4, \overline{\mathrm{a}} \cdot \overline{\mathrm{b}}=2$. If $\overline{\mathrm{c}}=(2 \overline{\mathrm{a}} \times \overline{\mathrm{b}})-3 \overline{\mathrm{~b}}$, then the angle between $\overline{\mathrm{b}}$ and $\overline{\mathrm{c}}$ is

If $\bar{a}, \bar{b}, \bar{c}, \bar{d}$ are unit vectors such that $\overline{\mathrm{a}} \cdot \overline{\mathrm{b}}=\frac{1}{2}, \overline{\mathrm{c}} \cdot \overline{\mathrm{d}}=\frac{1}{2}$ and the angle between $\overline{\mathrm{a}} \times \overline{\mathrm{b}}$ and $\overline{\mathrm{c}} \times \overline{\mathrm{d}}$ is $\frac{\pi}{6}$, then the value of $|[\overline{\mathrm{a}} \overline{\mathrm{b}} \overline{\mathrm{d}}] \overline{\mathrm{c}}-[\overline{\mathrm{a}} \overline{\mathrm{b}} \overline{\mathrm{c}}] \overline{\mathrm{d}}|=$

If $\bar{a}=4 \hat{i}+3 \hat{j}+\hat{k}, \bar{b}=\hat{i}-2 \hat{j}+2 \hat{k}$ then $\overline{\mathrm{a}} \times(\overline{\mathrm{a}} \times(\overline{\mathrm{a}} \times(\overline{\mathrm{a}} \times \overline{\mathrm{b}})))=$

If $\overline{\mathrm{a}}$ and $\overline{\mathrm{b}}$ are unit vectors and $\theta$ is the angle between them, then $\overline{\mathrm{a}}+\overline{\mathrm{b}}$ is a unit vector when $\theta$ is