Let $u$ and $v$ be two non-zero vectors. If $|u+v|=|u-v|$,then:

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

The angle $\theta$ between vectors $\vec{a} = \hat{i} + \hat{j} - \hat{k}$ and $\vec{b} = \hat{i} - \hat{j} + \hat{k}$ is . . . . . . .

$ABC$ is a right-angled triangle in which $\max \{AB, BC, AC\} = BC$. If the position vectors of $B$ and $C$ are respectively $3\hat{i}-2\hat{j}+\hat{k}$ and $5\hat{i}+\hat{j}-3\hat{k}$,then find the value of $AB \cdot AC + BA \cdot BC + CA \cdot CB$.

If $a, b$ and $c$ are three vectors such that $|a|=3, |b|=4$ and $|c|=5$ and $a+b+c=0$,then $a \cdot b$ is equal to

If the angle between $\vec{a}$ and $\vec{b}$ is $\frac{2\pi}{3}$ and the projection of $\vec{a}$ in the direction of $\vec{b}$ is $-2$,then find $|\vec{a}|$.