$\vec{a}, \vec{b}, \text{ and } \vec{c}$ are three vectors such that $|\vec{a}|=3, |\vec{b}|=5, |\vec{c}|=7$. If $\vec{a}, \vec{b}, \vec{c}$ are perpendicular to the vectors $\vec{b}+\vec{c}, \vec{c}+\vec{a}, \vec{a}+\vec{b}$ respectively,then $\sqrt{|\vec{a}+\vec{b}+\vec{c}|^2-2} = $

If $\overrightarrow{a}=-\hat{i}+\hat{j}+2 \hat{k}$, $\overrightarrow{b}=2 \hat{i}-\hat{j}-\hat{k}$ and $\overrightarrow{c}=-2 \hat{i}+\hat{j}+3 \hat{k}$, then the angle between $2 \overrightarrow{a}-\overrightarrow{c}$ and $\overrightarrow{a}+\overrightarrow{b}$ is

If $ai + 6j - k$ and $7i - 3j + 17k$ are perpendicular vectors,then the value of $a$ is

If $|\vec{a}|=\sqrt{26}$,$|\vec{b}|=7$ and $|\vec{a} \times \vec{b}|=35$,then $\vec{a} \cdot \vec{b}$ is-

Let $\overline{a}, \overline{b}$,and $\overline{c}$ be three non-zero vectors such that no two of these are collinear. If the vector $\overline{a}+2\overline{b}$ is collinear with $\overline{c}$ and $\overline{b}+3\overline{c}$ is collinear with $\overline{a}$,then $\overline{a}+2\overline{b}+6\overline{c}$ equals

Let $\overrightarrow{a} = \hat{i} + 2\hat{j} - \hat{k}$,$\overrightarrow{b} = \hat{i} - \hat{j}$ and $\overrightarrow{c} = \hat{i} - \hat{j} - \hat{k}$ be three given vectors. If $\overrightarrow{r}$ is a vector such that $\overrightarrow{r} \times \overrightarrow{a} = \overrightarrow{c} \times \overrightarrow{a}$ and $\overrightarrow{r} \cdot \overrightarrow{b} = 0$,then $\overrightarrow{r} \cdot \overrightarrow{a}$ is equal to ...........