In a quadrilateral $ABCD$, the point $P$ divides $DC$ in the ratio $1:3$ internally and $Q$ is the mid-point of $AC$. If $\vec{AB} + \vec{AD} + \vec{BC} - 2\vec{DC} = \lambda \vec{PQ}$, then the value of $\lambda$ is

If the vectors $x \hat{i}-3 \hat{j}+7 \hat{k}$ and $\hat{i}+y \hat{j}-z \hat{k}$ are collinear,then the value of $\frac{x y^2}{z}$ is equal to:

If $3 \hat{i}-5 \hat{j}+2 \hat{k}, 7 \hat{i}+2 \hat{j}-4 \hat{k}, \hat{i}-3 \hat{j}+4 \hat{k}$ and $-7 \hat{i}-17 \hat{j}+16 \hat{k}$ are position vectors of the points $A, B, C$ and $D$ respectively, then the angle between $\overrightarrow{AB}$ and $\overrightarrow{CD}$ is

$A$ vector $\vec{r}$ of magnitude $3\sqrt{2}$ units which makes angles of $\pi/4$ and $\pi/2$ respectively with $Y$ and $Z$ axes is:

If $\hat{i}-2 \hat{j}+3 \hat{k}$,$2 \hat{i}+3 \hat{j}+\hat{k}$,and $-3 \hat{i}-\hat{j}-2 \hat{k}$ are the position vectors of three points $A$,$B$,and $C$ respectively,then $A$,$B$,and $C$

For any two points $M$ and $N$ in the $XY$-plane, let $\overrightarrow{MN}$ denote the vector from $M$ to $N$, and $\overrightarrow{0}$ denote the zero vector. Let $P, Q$ and $R$ be three distinct points in the $XY$-plane. Let $S$ be a point inside the triangle $\triangle PQR$ such that $\overrightarrow{SP} + 5\overrightarrow{SQ} + 6\overrightarrow{SR} = \overrightarrow{0}$. Let $E$ and $F$ be the mid-points of the sides $PR$ and $QR$, respectively. Then the value of $\frac{\text{length of the line segment } EF}{\text{length of the line segment } ES}$ is: (in $.20$)