If $a, b$ and $c$ are mutually perpendicular vectors of the same magnitude,then the cosine of the angle between $a$ and $a+b+c$ is

If $p = i - 2j + 3k$ and $q = 3i + j + 2k,$ then a vector along $r$ which is a linear combination of $p$ and $q$ and also perpendicular to $q$ 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:

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

If $\bar{a}$ and $\bar{b}$ are two vectors such that $|\bar{a}|=5$, $|\bar{b}|=12$ and $|\bar{a}-\bar{b}|=13$, then $|2\bar{a}+\bar{b}|=$

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) $\vec{OG}$ | $(D)$ $\vec{SO}$ |
| | $(E)$ $\vec{OS}$ |