Forces of magnitudes $3$ and $2$ units acting in the directions $5\hat{i} + 3\hat{j} + 4\hat{k}$ and $3\hat{i} + 4\hat{j} - 5\hat{k}$ respectively act on a particle which is displaced from the points $(1, -1, -1)$ to $(3, 3, 1)$. The work done by the forces is equal to

If $\overrightarrow{a} \cdot \overrightarrow{b} = -|\overrightarrow{a}||\overrightarrow{b}|$,then the angle between $\overrightarrow{a}$ and $\overrightarrow{b}$ is (in $^{\circ}$)

The angle between the pair of lines with direction ratios $(1, 1, 2)$ and $(\sqrt{3} - 1, -\sqrt{3} - 1, 4)$ is ......... $^o$.

If $\vec{a}, \vec{b}$ and $\vec{c}$ are unit vectors such that $\vec{a}$ is perpendicular to $\vec{b}$ and $\vec{c}$,and $|\vec{a} + \vec{b} + \vec{c}| = 1$,then the angle between $\vec{b}$ and $\vec{c}$ is:

If the angle between two vectors $\vec{u} = i + k$ and $\vec{v} = i - j + ak$ is $\pi / 3,$ then the value of $a$ is:

If $\overline{a}, \overline{b}, \overline{c}$ are non-coplanar vectors and $\overline{p}=\frac{\overline{b} \times \overline{c}}{[\overline{a} \overline{b} \overline{c}]}, \overline{q}=\frac{\overline{c} \times \overline{a}}{[\overline{a} \overline{b} \overline{c}]}, \overline{r}=\frac{\overline{a} \times \overline{b}}{[\overline{a} \overline{b} \overline{c}]}, \quad$ then $2 \overline{a} \cdot \overline{p}+\overline{b} \cdot \overline{q}+\overline{c} \cdot \overline{r}=$