If $\overline{p}=2 \hat{i}+\hat{k}$,$\overline{q}=\hat{i}+\hat{j}+\hat{k}$,$\overline{r}=4 \hat{i}-3 \hat{j}+7 \hat{k}$ and a vector $\overline{m}$ is such that $\overline{m} \times \overline{q}=\overline{r} \times \overline{q}$ and $\overline{m} \cdot \overline{p}=0$,then $\overline{m} = \dots$

If $\overrightarrow{a}$ and $\overrightarrow{b}$ are unit vectors,then the greatest value of $\sqrt{3}|\overrightarrow{a}+\overrightarrow{b}|+|\overrightarrow{a}-\overrightarrow{b}|$ is

If $\vec{a}=\hat{i}+(\tan \theta) \hat{j}+\left(\frac{3}{\sqrt{\sin \frac{\theta}{2}}}\right) \hat{k}$ and $\vec{b}=\tan \theta(\hat{j}-\hat{i})-\left(2 \sqrt{\sin \frac{\theta}{2}}\right) \hat{k}$ are orthogonal vectors and $\vec{c}=(\sin 2 \theta) \hat{i}-2 \hat{j}+2 \hat{k}$ makes an obtuse angle with $X$-axis, then $\theta=$

Let $b = 4i + 3j$ and $c$ be two vectors perpendicular to each other in the $xy$-plane. All vectors in the same plane having projections $1$ and $2$ along $b$ and $c$ respectively,are given by

Let $a = i + 2j + k$,$b = i - j + k$,$c = i + j - k$. $A$ vector in the plane of $a$ and $b$ has projection $\frac{1}{\sqrt{3}}$ on $c$. Then,one such vector is

If $S$ is the circumcentre, $O$ is the orthocentre and $G$ is the centroid of a triangle $ABC$, then match the items of the List-$I$ with those of the items of List-$II$ given below.
| List-$I$ | List-$II$ |
| :--- | :--- |
| $(i)$ $\vec{SA} + \vec{SB} + \vec{SC}$ | $(A)$ $2\vec{OS}$ |
| (ii) $\vec{GA} + \vec{GB} + \vec{GC}$ | $(B)$ $\frac{2}{3}\vec{OS}$ |
| (iii) $\vec{OA} + \vec{OB} + \vec{OC}$ | $(C)$ $\vec{0}$ |
| (iv) $\vec{OG}$ | $(D)$ $\vec{SO}$ |
| | $(E)$ $\vec{OS}$ |