If $|a| = 2$,$|b| = 5$ and $|a \times b| = 8$,then $a \cdot b$ is equal to

Prove that $(\vec{a}+\vec{b}) \cdot(\vec{a}+\vec{b})=|\vec{a}|^{2}+|\vec{b}|^{2},$ if and only if $\vec{a}$ and $\vec{b}$ are perpendicular,given $\vec{a} \neq \vec{0}, \vec{b} \neq \vec{0}.$

If the position vectors of the vertices $A, B$,and $C$ of a triangle $ABC$ are $4\hat{i} + 7\hat{j} + 8\hat{k}$,$2\hat{i} + 3\hat{j} + 4\hat{k}$,and $2\hat{i} + 5\hat{j} + 7\hat{k}$ respectively,then the position vector of the point where the bisector of angle $A$ meets $BC$ is:

If $a, b, c$ are unit vectors satisfying the relation $a+b+\sqrt{3} c=0$, then the angle between $a$ and $b$ is

Let $O$ be the origin and let $PQR$ be an arbitrary triangle. The point $S$ is such that $\overline{OP} \cdot \overline{OQ} + \overline{OR} \cdot \overline{OS} = \overline{OR} \cdot \overline{OP} + \overline{OQ} \cdot \overline{OS} = \overline{OQ} \cdot \overline{OR} + \overline{OP} \cdot \overline{OS}$. Then the triangle $PQR$ has $S$ as its

Magnitudes of vectors $\vec a, \vec b, \vec c$ are $3, 4, 5$ respectively. If $\vec a$ and $\vec b + \vec c$,$\vec b$ and $\vec c + \vec a$,and $\vec c$ and $\vec a + \vec b$ are mutually perpendicular,then find the magnitude of $|\vec a + \vec b + \vec c|$.