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 $\bar{a}$ and $\bar{b}$ are two vectors such that $|\bar{a}|=|\bar{b}|=\sqrt{6}$ and $\bar{a} \cdot \bar{b}=-1$, then find the value of $|\bar{a} \times \bar{b}| \sin(\theta)$, where $\theta$ is the angle between $\bar{a}$ and $\bar{b}$.

If the angle between unit vectors $\vec{a}$ and $\vec{b}$ is $2\theta$,and $|\vec{a} - \vec{b}| < 1$ with $0 \le \theta \le \pi$,then in which interval does $\theta$ lie?

If the vertices $A, B, C$ of a triangle $ABC$ are $(1,2,3), (-1,0,0), (0,1,2)$ respectively,then find $\angle ABC$. $[\angle ABC \text{ is the angle between the vectors } \overrightarrow{BA} \text{ and } \overrightarrow{BC}]$.

If $\overline{a}, \overline{b}, \overline{c}$ are three vectors such that $\overline{a} \cdot(\overline{b}+\overline{c})+\overline{b} \cdot(\overline{c}+\overline{a})+\overline{c} \cdot(\overline{a}+\overline{b})=0$ and $|\overline{a}|=1$,$|\overline{b}|=8$ and $|\overline{c}|=4$,then $|\overline{a}+\overline{b}+\overline{c}|$ has the value

The points $O, A, B, C, D$ are such that $\overrightarrow{OA} = a, \overrightarrow{OB} = b, \overrightarrow{OC} = 2a + 3b$ and $\overrightarrow{OD} = a - 2b$. If $|a| = 3|b|$,then the angle between $\overrightarrow{BD}$ and $\overrightarrow{AC}$ is