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

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 $.20$)

If $ABCD$ is a parallelogram,$\overrightarrow{AB} = 2i + 4j - 5k$ and $\overrightarrow{AD} = i + 2j + 3k,$ then the unit vector in the direction of $\overrightarrow{BD}$ is

Let $\vec{OA} = -4\hat{i} + 3\hat{k}$ and $\vec{OB} = 14\hat{i} + 2\hat{j} - 5\hat{k}$. If $\vec{OD}$ bisects $\angle AOB$ and $|\vec{OD}| = \sqrt{6}$,then $\vec{OD} =$

The points having position vectors $2i + 3j + 4k,$ $3i + 4j + 2k,$ and $4i + 2j + 3k$ are the vertices of:

The position vector of the point of intersection of the medians (centroid) of a triangle,whose vertices are $A(1, 2, 3)$,$B(1, 0, 3)$,and $C(4, 1, -3)$ is