$PQRS$ is a quadrilateral and $PQ=a, QR=b, SP=a-b$. $M$ is the mid-point of $QR$ and $X$ is a point on $SM$ such that $SX=\frac{4}{5}SM$. If $SM=m(4a-b)$ and $SX=n(4a-b)$,then $m+n=$

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:

If $a$ is collinear with $b = 3 \hat{i} + 6 \hat{j} + 6 \hat{k}$ and $a \cdot b = 27$,then $|a| =$

If the position vectors of two points $A$ and $B$ are $\vec{a} + 3\vec{b}$ and $\vec{a} - 2\vec{b}$ respectively,find the position vector of the point that divides $AB$ in the ratio $2:5$.

If $\vec{r}_1 = 2\hat{i} + 4\hat{j} - 5\hat{k}$ and $\vec{r}_2 = \hat{i} + 2\hat{j} + 3\hat{k}$,find the unit vector parallel to the resultant vector of $\vec{r}_1$ and $\vec{r}_2$.

If $\bar{x}$ is a non-zero vector and $k > 0, k \neq 1$,then $\frac{-k \bar{x}}{|\bar{x}|}$ is $.......$ .