For a triangle $ABC$,if $\vec{AB} = 3\hat{i} + 4\hat{k}$ and $\vec{AC} = 5\hat{i} - 2\hat{j} + 4\hat{k}$,then the length of the median drawn from vertex $A$ is:

If $a$ and $b$ are two vectors such that $a \cdot b = 0$ and $a \times b = 0$,then:

$A$ vector $\vec{a}$ has components $2p$ and $1$ with respect to a rectangular Cartesian system. The system is rotated through a certain angle about the origin in the anti-clockwise sense. If $\vec{a}$ has components $p+1$ and $1$ with respect to the new system,then:

The vectors $\vec{AB} = 3\hat{i} + 4\hat{k}$ and $\vec{AC} = 5\hat{i} - 2\hat{j} + 4\hat{k}$ are the sides of a $\triangle ABC$. The length of the median through $A$ is:

Given $p = 2a - 3b$,$q = a - 2b + c$,and $r = -3a + b + 2c$,where $a, b,$ and $c$ are non-zero,non-coplanar vectors,then the vector $-2a + 3b - c$ is equal to:

For a parallelogram $ABCD$,if $L$ and $M$ are mid-points of $BC$ and $CD$ respectively,then $AL + AM =$