$\overline{a}, \overline{b}$ and $\overline{c}$ are three vectors such that $\overline{a}+\overline{b}+\overline{c}=\overline{0}$ and $|\overline{a}|=3, |\overline{b}|=5, |\overline{c}|=7$. The angle between $\overline{a}$ and $\overline{b}$ is:

If $|a \times b| = 4$ and $|a \cdot b| = 2$,then $|a|^2 |b|^2 = $

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:

$|a \times b|^2 + (a \cdot b)^2 = ?$

Let $\vec{a}, \vec{b}$ and $\vec{c}$ be three unit vectors such that $|\vec{a}-\vec{b}|^{2}+|\vec{a}-\vec{c}|^{2}=8$. Then $|\vec{a}+2\vec{b}|^{2}+|\vec{a}+2\vec{c}|^{2}$ is equal to

$A$ unit vector in the $xy$-plane that makes an angle of $45^{\circ}$ with the vector $(i + j)$ and an angle of $60^{\circ}$ with the vector $(3i - 4j)$ is: