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

If $i + 2j + 3k$ is parallel to the sum of the vectors $3i + \lambda j + 2k$ and $-2i + 3j + k$,then $\lambda = \dots$

If $\vec{a}$ is a non-zero vector of modulus $|\vec{a}|$ and $m$ is a non-zero scalar,then $m\vec{a}$ is a unit vector if:

If $\bar{a}, \bar{b}, \bar{c}$ are non-zero non-collinear vectors and $\bar{a} \times \bar{b} = \bar{b} \times \bar{c} = \bar{c} \times \bar{a}$,then $\bar{a} + \bar{b} + \bar{c} = $

If $\vec{a}=\hat{i}+\hat{j}+\hat{k}, \vec{b}=2 \hat{i}-\hat{j}+3 \hat{k}$ and $\vec{c}=\hat{i}-2 \hat{j}+\hat{k},$ find a unit vector parallel to the vector $2 \vec{a}-\vec{b}+3 \vec{c}.$

$P$ and $Q$ are the points of trisection of the line segment $AB$. If $2 \hat{i}-5 \hat{j}+3 \hat{k}$ and $4 \hat{i}+\hat{j}-6 \hat{k}$ are the position vectors of $A$ and $B$ respectively, then the position vector of the point which divides $PQ$ in the ratio $2:3$ is