$A$ unit vector in $XY$-plane making an angle $45^{\circ}$ with $\hat{i}+\hat{j}$ and an angle $60^{\circ}$ with $3\hat{i}-4\hat{j}$ is

$ABCD$ is a tetrahedron. $\bar{i}-2\bar{j}+3\bar{k}$, $-2\bar{i}+\bar{j}+3\bar{k}$, and $3\bar{i}+2\bar{j}-\bar{k}$ are the position vectors of the points $A, B, C$ respectively. $-\bar{i}+2\bar{j}-3\bar{k}$ is the position vector of the centroid of the triangular face $BCD$. If $G$ is the centroid of the tetrahedron, then $GD=$

If $G$ is the centroid of the $\triangle ABC$,then $\vec{GA} + \vec{GB} + \vec{GC}$ is equal to

In $\triangle ABC$,$L, M, N$ are points on $BC, CA, AB$ respectively,dividing them in the ratio $1:2, 2:3, 3:5$. If the point $K$ divides $AB$ in the ratio $5:3$,then find the value of $\left| \frac{\vec{AL} + \vec{BM} + \vec{CN}}{\vec{CK}} \right|$.

For any vector $\vec{a} = a_1 \hat{i} + a_2 \hat{j} + a_3 \hat{k}$,with $10|a_i| < 1$,$i = 1, 2, 3$,consider the following statements:
$(A): \max \{|a_1|, |a_2|, |a_3|\} \leq |\vec{a}|$
$(B): |\vec{a}| \leq 3 \max \{|a_1|, |a_2|, |a_3|\}$

Find the unit vector in the direction of vector $\overrightarrow{PQ},$ where $P$ and $Q$ are the points $(1, 2, 3)$ and $(4, 5, 6),$ respectively.