If $A(4,7,8)$,$B(2,3,4)$,and $C(2,5,7)$ are the position vectors of the vertices of a triangle $ABC$ and if the internal bisector of $\angle A$ meets $BC$ at $D$,then $AD=$

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:

Let $\vec{u}$ and $\vec{v}$ be two non-zero vectors with the intermediate angle $45^{\circ}$. Then $|\vec{u} \times \vec{v}|=$

The projection of $\overline{AB}$ on $\overline{CD}$,where $A \equiv (2, -3, 0)$,$B \equiv (1, -4, -2)$,$C \equiv (4, 6, 8)$,and $D \equiv (7, 0, 10)$ is

Let $a, b$ and $c$ be three non-coplanar vectors. The vector equation of a line which passes through the point of intersection of two lines,one joining the points $a+2b-5c$ and $-a-2b-3c$,and the other joining the points $-4c$ and $6a-4b+4c$,is

If the resultant of two forces is of magnitude $P$ and equal to one of them and perpendicular to it,then the other force is