If $|\vec{a}|=13, |\vec{b}|=5$ and $\vec{a} \cdot \vec{b}=60$,then $|\vec{a} \times \vec{b}|=$

If $\hat{a}, \hat{b}, \hat{c}$ are unit vectors,then the least value of $|\hat{a}+\hat{b}|^2+|\hat{b}+\hat{c}|^2+|\hat{c}+\hat{a}|^2$ will be-

Let $\vec{b}$ and $\vec{c}$ be non-collinear vectors satisfying $\vec{a} \times (\vec{b} \times \vec{c}) + (\vec{a} \cdot \vec{b})\vec{b} = (4 - 2x - \sin y)\vec{b} + (x^2 - 1)\vec{c}$ and $(\vec{c} \cdot \vec{c})\vec{a} = \vec{c}$,then $x$ is equal to

If $a, b$ and $c$ are three non-zero vectors,no two of which are collinear. If the vector $a + 2b$ is collinear with $c$ and $b + 3c$ is collinear with $a$,then ($\lambda$ being some non-zero scalar) $a + 2b + 6c$ is equal to

The cosine of the angle $A$ of the triangle with vertices $A(1, -1, 2)$,$B(6, 11, 2)$,and $C(1, 2, 6)$ is:

$\bar{a}, \bar{b}, \bar{c}$ are three non-coplanar and mutually perpendicular vectors of same magnitude $K$. If $\bar{r}$ is any vector satisfying $\bar{a} \times ((\bar{r}-\bar{b}) \times \bar{a}) + \bar{b} \times ((\bar{r}-\bar{c}) \times \bar{b}) + \bar{c} \times ((\bar{r}-\bar{a}) \times \bar{c}) = \bar{0}$, then $\bar{r} =$