If the vectors $\vec{AB} = \hat{i} + 3\hat{j} + 4\hat{k}$ and $\vec{AC} = 5\hat{i} + \hat{j} + 2\hat{k}$ are two sides of a triangle $ABC$,whose centroid is $G$,then $|\vec{AG}| = $

If $\vec{a}$ and $\vec{b}$ are two non-collinear vectors,then $|\vec{b}| \vec{a} + |\vec{a}| \vec{b}$ represents:

If $\overline{AB} \parallel \overline{CD}$ and the position vectors of the four points $A, B, C,$ and $D$ are $2i + j, i - 3j, 3i + 2j,$ and $i + \lambda j$ respectively,then the value of $\lambda$ is .....

If $\vec{AB} = 3 \hat{i} + 5 \hat{j} + 4 \hat{k}$ and $\vec{AC} = 5 \hat{i} - 5 \hat{j} + 2 \hat{k}$ represent the sides of triangle $ABC$,then the length of the median through $A$ is

$R$ divides the line joining two points $P$ and $Q$ whose position vectors are $\hat{i}+2 \hat{j}-\hat{k}$ and $-\hat{i}+\hat{j}+\hat{k}$ respectively in the ratio $2: 1$ externally. $S$ divides $PQ$ internally in the ratio $2: 1$. Then,the position vector of the midpoint of the line joining $R$ and $S$ is

If the position vectors of the points $A$ and $B$ are $\vec{a} = \hat{i} + 3\hat{j} - \hat{k}$ and $\vec{b} = 3\hat{i} - \hat{j} - 3\hat{k}$,then what will be the position vector of the midpoint of $AB$?