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

If $\vec{a}=\hat{i}+\hat{j}+\hat{k}$,$\vec{b}=\hat{i}-\hat{j}+\hat{k}$,and $\vec{c}=\hat{i}-\hat{j}-\hat{k}$ are three vectors,then the vector $\vec{r}$ in the plane of $\vec{a}$ and $\vec{b}$,whose projection on $\vec{c}$ is $\frac{1}{\sqrt{3}}$,is given by:

Let $u = -2 \hat{i} + 2 \hat{j} + \hat{k}$ and $v = \hat{i} - 2 \hat{j} + 2 \hat{k}$. Then the component of $v$ on $u$ is

The projection of $\overrightarrow{a} = 3 \hat{i} - \hat{j} + 5 \hat{k}$ on $\overrightarrow{b} = 2 \hat{i} + 3 \hat{j} + \hat{k}$ is

Let $\overrightarrow{OA}$ and $\overrightarrow{OB}$ be two sides of a triangle. The median $\overrightarrow{AM}$ is perpendicular to the angle bisector $\overrightarrow{OL}$ and $|\overrightarrow{AM}|:|\overrightarrow{OL}|=1:2$. The angle between $\overrightarrow{OA}$ and $\overrightarrow{OB}$ is

If the vectors $a\,i - 2j + 3k$ and $3i + 6j - 5k$ are perpendicular to each other,then $a$ is given by