If $\bar{a}$ and $\bar{b}$ are unit vectors and $\theta$ is the angle between them,then $\bar{a}+\bar{b}$ is a unit vector when $\theta$ is

Let $\bar{a} = \bar{i} + 2\bar{j} + 3\bar{k}$,$\bar{b} = 2\bar{i} - 3\bar{j} + \bar{k}$,and $\bar{c} = 3\bar{i} + \bar{j} - 2\bar{k}$ be three vectors. If $\bar{r}$ is a vector such that $\bar{r} \cdot \bar{a} = 0$,$\bar{r} \cdot \bar{b} = -2$,and $\bar{r} \cdot \bar{c} = 6$,then find the value of $\bar{r} \cdot (3\bar{i} + \bar{j} + \bar{k})$.

If in a right-angled triangle $ABC$,the hypotenuse $|\overrightarrow{AB}| = p$,then $\overrightarrow{AB} \cdot \overrightarrow{AC} + \overrightarrow{BC} \cdot \overrightarrow{BA} + \overrightarrow{CA} \cdot \overrightarrow{CB} = $

Let $\vec{\alpha}, \vec{\beta}, \vec{\gamma}$ be three non-zero vectors which are pairwise non-collinear. If $\vec{\alpha}+3 \vec{\beta}$ is collinear with $\vec{\gamma}$ and $\vec{\beta}+2 \vec{\gamma}$ is collinear with $\vec{\alpha}$,then $\vec{\alpha}+3 \vec{\beta}+6 \vec{\gamma}$ is

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

$A$ vector of magnitude $\sqrt{51}$ which makes equal angles with the vectors $\bar{a}=\frac{1}{3}(\bar{i}-2 \bar{j}+2 \bar{k})$,$\bar{b}=\frac{1}{5}(-4 \bar{i}-3 \bar{k})$ and $\bar{c}=\bar{j}$,is