If $\theta$ is an obtuse angle between vectors $\overline{a}$ and $\overline{b}$ such that $|\overline{a}|=5$,$|\overline{b}|=3$ and $|\overline{a} \times \overline{b}|=5 \sqrt{5}$,then $\overline{a} \cdot \overline{b}=$

If $\vec{a}=\hat{i}+\hat{j}+\hat{k}$ and $\vec{b}=\hat{i}+2\hat{j}+3\hat{k}$,then $(\vec{a}+\vec{b}) \cdot (\vec{a}-\vec{b}) = $ . . . . . . .

If $\vec{a}$ and $\vec{b}$ are two unit vectors with $(\vec{a}, \vec{b}) = \theta$ and $|\vec{a} - \vec{b}| = 1$,then $2|\vec{a} + \vec{b}| \cos \frac{\theta}{2} =$

The angle between two diagonals of a cube is:

$A(\vec{a}), B(\vec{b}), C(\vec{c}), D(\vec{d})$ are four concyclic points, such that $x \vec{a}+y \vec{b}+z \vec{c}+t \vec{d}=\vec{0}$ and $x+y+z+t=0$, where $x, y, z, t$ are constants not all zero. If the chords $AB$ and $CD$ intersect at $P$, then:

The angle between the vectors $a + b$ and $a - b$,when $a = (1, 1, 4)$ and $b = (1, -1, 4)$ is .............. $^o$