Let $\vec{a}=\hat{i}-3 \hat{j}+7 \hat{k}$,$\vec{b}=2 \hat{i}-\hat{j}+\hat{k}$ and $\vec{c}$ be a vector such that $(\vec{a}+2 \vec{b}) \times \vec{c}=3(\vec{c} \times \vec{a})$. If $\vec{a} \cdot \vec{c}=130$,then $\vec{b} \cdot \vec{c}$ is equal to ....................

If $\alpha=\hat{i}-3 \hat{j}$ and $\beta=\hat{i}+2 \hat{j}-\hat{k}$,express $\beta$ in the form $\beta=\beta_1+\beta_2$,where $\beta_1$ is parallel to $\alpha$ and $\beta_2$ is perpendicular to $\alpha$. Then $\beta_1$ is given by:

Two forces $\vec{F_1} = 2\hat{i} - 5\hat{j} + 6\hat{k}$ and $\vec{F_2} = -\hat{i} + 2\hat{j} - \hat{k}$ act on a particle. The particle is displaced from point $P(4\hat{i} - 3\hat{j} + 2\hat{k})$ to point $Q(6\hat{i} + \hat{j} + 3\hat{k})$. The work done by the forces is ............. units.

Given vertices $A(3, -1)$,$B(2, 3)$,and $C(5, 1)$,find $m \angle A$.

If $|\vec{a}|=13, |\vec{b}|=5$ and $\vec{a} \cdot \vec{b}=60$,then $|\vec{a} \times \vec{b}|=$

If $\overline{a}$ and $\overline{b}$ are two unit vectors such that $\overline{a}+2\overline{b}$ and $5\overline{a}-4\overline{b}$ are perpendicular to each other,then the angle between $\overline{a}$ and $\overline{b}$ is