The magnitude of the sum of vectors $(1, -\sqrt{2})$ and $(2, \sqrt{2})$ is $\ldots$

$\vec{a}, \vec{b}, \vec{c}$ are non-coplanar vectors. If $\alpha \vec{d}=\vec{a}+\vec{b}+\vec{c}$ and $\beta \vec{a}=\vec{b}+\vec{c}+\vec{d}$,then $|\vec{a}+\vec{b}+\vec{c}+\vec{d}|=$

If $\hat{i}$ is the position vector of the centroid $G$ of triangle $ABC$ and $2\hat{i}+\hat{j}+\hat{k}$ and $2\hat{i}+4\hat{j}-4\hat{k}$ are respectively the position vectors of its vertices $A$ and $B$,then $AG^2+BG^2+CG^2=$

$A$ unit vector $\vec{a}$ makes an angle $\frac{\pi}{4}$ with the $z$-axis. If $\vec{a} + \hat{i} + \hat{j}$ is a unit vector,then $\vec{a}$ is equal to

Let $\vec{i}+\vec{j}+\vec{k}$,$a_1 \vec{i}+b_1 \vec{j}+c_1 \vec{k}$,$a_2 \vec{i}+b_2 \vec{j}+c_2 \vec{k}$,and $a_3 \vec{i}+b_3 \vec{j}+c_3 \vec{k}$ be the position vectors of the points $A, B, C, D$ respectively. The position vector of the centroid of the triangular face $BCD$ is $\frac{2}{3}(\vec{i}+\vec{j}+\vec{k})$. If $\alpha \vec{i}+\beta \vec{j}+\gamma \vec{k}$ is the position vector of the centroid of the tetrahedron $ABCD$,then find the value of $2 \alpha+\beta+\gamma$.

The position vectors of $P$ and $Q$ are respectively $\overrightarrow{a}$ and $\overrightarrow{b}$. If $R$ is a point on the line $PQ$ such that $\overrightarrow{PR}=5 \overrightarrow{PQ}$,then the position vector of $R$ is