The set of all real values of $c$ such that the angle between the vectors $\vec{a} = cx \hat{i} - 6 \hat{j} + 3 \hat{k}$ and $\vec{b} = x \hat{i} + 2 \hat{j} + 2cx \hat{k}$ is an obtuse angle for all real $x$ is:

The points $O, A, B, C, D$ are such that $\overrightarrow{OA} = \vec{a}$,$\overrightarrow{OB} = \vec{b}$,$\overrightarrow{OC} = 2\vec{a} + 3\vec{b}$,and $\overrightarrow{OD} = \vec{a} - 2\vec{b}$. If $|\vec{a}| = 3|\vec{b}|$,then the angle between $\overrightarrow{BD}$ and $\overrightarrow{AC}$ is:

$OA$ and $OB$ are two vectors of magnitudes $5$ and $6$ respectively. If $\angle BOA = 60^{\circ}$,then $OA \cdot OB$ is equal to

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 $2 \overline{a} \cdot \overline{p}+\overline{b} \cdot \overline{q}+\overline{c} \cdot \overline{r}=$

Let $ABC$ be an acute scalene triangle,and $O$ and $H$ be its circumcentre and orthocentre respectively. Further,let $N$ be the mid-point of $OH$. The value of the vector sum $\overrightarrow{NA}+\overrightarrow{NB}+\overrightarrow{NC}$ is

Let $\vec{a}, \vec{b}$ and $\vec{c}$ be three vectors such that $|\vec{a}|=3, |\vec{b}|=4, |\vec{c}|=5$ and each one of them is perpendicular to the sum of the other two. Find $|\vec{a}+\vec{b}+\vec{c}|$.