The non-zero vectors $\vec{a}$,$\vec{b}$,and $\vec{c}$ are related by $\vec{a} = 8\vec{b}$ and $\vec{c} = -7\vec{b}$. Then the angle between $\vec{a}$ and $\vec{c}$ is ............... $^\circ $

Find a vector of magnitude $11$ in the direction opposite to that of $\overrightarrow{PQ}$,where $P$ and $Q$ are the points $(1,3,2)$ and $(-1,0,8)$ respectively.

If $G$ and $G'$ are the centroids of the triangles $ABC$ and $A'B'C'$ respectively,then $\overrightarrow{AA'} + \overrightarrow{BB'} + \overrightarrow{CC'} = $

If $P$ and $Q$ are two points on the curve $y = 2^{x+2}$ in the rectangular Cartesian coordinate system such that $\overline{OP} \cdot \hat{i} = -1$ and $\overline{OQ} \cdot \hat{i} = 2$,then $\overline{OQ} - 4\overline{OP} = $

The value of $x$ if $x(\hat{i}+\hat{j}+\hat{k})$ is a unit vector is:

The position vectors of four points $P, Q, R, S$ are $2a + 4c$,$5a + 3\sqrt{3}b + 4c$,$-2\sqrt{3}b + c$,and $2a + c$ respectively. Then: