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

If $\vec{a} = \hat{i} + \hat{j} - 2 \hat{k}$,$\vec{b} = 2 \hat{i} - \hat{j} + \hat{k}$ and $\vec{c} = 3 \hat{i} - \hat{k}$. If $\vec{c} = m \vec{a} + n \vec{b}$ then $m + n = $

$A$ vector $\vec{r}$ is inclined at equal angles to the three axes. If the magnitude of $\vec{r}$ is $2 \sqrt{3}$ units,find $\vec{r}$.

The position vectors of the points $P$ and $Q$ are respectively $-2 \bar{i}-3 \bar{j}+\bar{k}$ and $3 \bar{i}+3 \bar{j}+2 \bar{k}$. The ratio in which the point having position vector $\frac{-9}{2} \bar{i}-6 \bar{j}+\frac{1}{2} \bar{k}$ divides the line segment joining $P$ and $Q$ is

If $A(2 \hat{i} + \hat{j} - \hat{k})$, $B(\lambda \hat{i} + 5 \hat{j} + 4 \hat{k})$, $C(-4 \hat{i} + 3 \hat{j} + 2 \hat{k})$ and $D(-\hat{i} - 2 \hat{j} + 3 \hat{k})$ are four points in space such that $\overrightarrow{AB} = x \overrightarrow{AC} + y \overrightarrow{AD}$ for some real numbers $x \neq 0, y \neq 0$, then $17(\lambda + 9) =$ ?

If the sum of two unit vectors is a unit vector, then the magnitude of their difference is