If $a = \hat{i} + 2 \hat{j} + 3 \hat{k}$,$b = 2 \hat{i} + 3 \hat{j} + \hat{k}$,$c = 8 \hat{i} + 13 \hat{j} + 9 \hat{k}$ and $x a + y b + z c = 0$,then $\frac{x y}{z^2} =$

Let $P, Q, R,$ and $S$ be points on a plane with position vectors $-2\hat{i} - \hat{j}$,$4\hat{i}$,$3\hat{i} + 3\hat{j}$,and $-3\hat{i} + 2\hat{j}$ respectively. What type of quadrilateral is $PQRS$?

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 $\hat{i}+2 \hat{j}+3 \hat{k}$ and $3 \hat{i}+2 \hat{j}+\hat{k}$ are sides of a parallelogram,then a unit vector parallel to one of the diagonals of the parallelogram is

The position vector of the point which divides the line segment joining the points with position vectors $2a - 3b$ and $3a - 2b$ internally in the ratio $2 : 3$ is:

The figure formed by the four points $(\hat{i}+\hat{j}-\hat{k}), (2\hat{i}+3\hat{j}), (5\hat{j}-2\hat{k})$ and $(\hat{k}-\hat{j})$ is