If $|\vec{a} \cdot \vec{b}| = |\vec{a} \times \vec{b}|$, $\vec{a} \cdot \vec{b} < 0$ and $\theta$ is the angle between $\vec{a}$ and $\vec{b}$, then the value of $\sin \theta + \tan \theta$ is...

In a plane, $\vec{a}$ and $\vec{b}$ are the position vectors of two points $A$ and $B$ respectively. $A$ point $P$ with position vector $\vec{r}$ moves on that plane in such a way that $|\vec{r}-\vec{a}| - |\vec{r}-\vec{b}| = c$ (where $c$ is a real constant). The locus of $P$ is a conic section whose eccentricity is:

Given three vectors $\bar{a}, \bar{b}, \bar{c}$,two of which are collinear. If $\bar{a}+\bar{b}$ is collinear with $\bar{c}$ and $\bar{b}+\bar{c}$ is collinear with $\bar{a}$,and $|\bar{a}|=|\bar{b}|=|\bar{c}|=\sqrt{2}$,then $\bar{a} \cdot \bar{b}+\bar{b} \cdot \bar{c}+\bar{c} \cdot \bar{a}=$

Let $x \in R$ and $\log_2 x > 0$. Then,the vectors $A = (2, \log_2 x, s)$ and $B = (\log_2 x, s, \log_2 x)$ include an acute angle if

If the lengths of three vectors $\bar{a}, \bar{b}$ and $\bar{c}$ are $5, 12, 13$ units respectively,and each one is perpendicular to the sum of the other two,then $|\bar{a}+\bar{b}+\bar{c}| = \dots$

Let $\vec{a}=2 \hat{i}+3 \hat{j}+\hat{k}$,$\vec{b}=4 \hat{i}+\hat{j}$,$\vec{c}=\hat{i}-3 \hat{j}-7 \hat{k}$. If $\vec{r}=x \hat{i}+y \hat{j}+z \hat{k}$,$\vec{r} \cdot \vec{a}=9$,$\vec{r} \cdot \vec{b}=7$,$\vec{r} \cdot \vec{c}=6$,then $(x, y, z) = $