If $\theta$ is the angle between two vectors $\vec{a}$ and $\vec{b},$ then $\vec{a} \cdot \vec{b} \geq 0$ only when

If $|a| = 3, |b| = 4, |c| = 5$ and $a + b + c = 0,$ then the angle between $a$ and $b$ is

Suppose $\vec{a}+\vec{b}+\vec{c}=0$,$|\vec{a}|=3$,$|\vec{b}|=5$,$|\vec{c}|=7$. Then the angle between $\vec{a}$ and $\vec{b}$ is:

If $\bar{a}, \bar{b}$ and $\bar{c}$ are vectors such that $|\bar{a}| = |\frac{\bar{b}}{2}| = |\frac{\bar{c}}{3}| = 1$; $\bar{b}$ and $\bar{c}$ are perpendicular; and the projections of $\bar{b}$ and $\bar{c}$ on $\bar{a}$ are equal,then $|\bar{a} - \bar{b} + \bar{c}| = $

If $a$ and $b$ are unit vectors and $\alpha$ is the angle between them, then $a+b$ is a unit vector when $\cos \alpha=$

Coordinate planes and the planes $\pi_1, \pi_2, \pi_3$ which are respectively parallel to $YZ, ZX, XY$ planes at distances $a, b, c$,form a rectangular parallelepiped. $d_1$ is a diagonal of the face on the $XY$-plane not passing through the origin,and $d_2$ is a diagonal of plane $\pi_2$ coterminous with $d_1$. If none of the coordinates of the vertices of the parallelepiped are negative and the angle between $d_1$ and $d_2$ is $\theta$,then $\cos \theta=$