The position vectors of $A$ and $B$ are $(\hat{i}+\hat{j}+\hat{k})$ and $(\frac{1}{3} \hat{j}+\frac{1}{3} \hat{k})$. If $B$ divides the line segment $AC$ in the ratio $2:1$,then the position vector of $C$ is

In a regular hexagon $ABCDEF$,$AD + EB + FC = (3\lambda - 8) AB$. Then $\lambda =$

The projection of the vector $i+j+k$ along the vector $j$ is

Given $\overrightarrow{p} = 3 \hat{i} + 2 \hat{j} + 4 \hat{k}$,$\overrightarrow{a} = \hat{i} + \hat{j}$,$\overrightarrow{b} = \hat{j} + \hat{k}$,$\overrightarrow{c} = \hat{i} + \hat{k}$ and $\overrightarrow{p} = x \overrightarrow{a} + y \overrightarrow{b} + z \overrightarrow{c}$,then $x, y, z$ are respectively:

The system of unit vectors $i, j, k$ is

If $M_1, M_2, M_3$ and $M_4$ are respectively the magnitudes of the vectors $\vec{a}_1 = 2\hat{i} - \hat{j} + \hat{k}$,$\vec{a}_2 = -3\hat{i} - 4\hat{j} - 4\hat{k}$,$\vec{a}_3 = -\hat{i} + \hat{j} - \hat{k}$,and $\vec{a}_4 = -\hat{i} + 3\hat{j} + \hat{k}$,then the correct order of $M_1, M_2, M_3$ and $M_4$ is: