The direction of the vectors $(1, 1, 2)$ and $(2, 1, 0)$ is $.......$

$ABCDE$ is a pentagon. Forces $\overline{AB}, \overline{AE}, \overline{DC}, \overline{ED}$ act at a point. Which force should be added to this system to make the resultant $2\overline{AC}$?

If $2 \hat{i}-\hat{j}+\hat{k}$ and $\hat{i}-3 \hat{j}-5 \hat{k}$ are the position vectors of the points $A$ and $B$ respectively, $C$ divides $AB$ in the ratio $2:3$ and $M$ is the mid-point of $AB$, then $5(\text{position vector of } C) - 2(\text{position vector of } M) =$

Let the position vectors of the vertices $A, B$ and $C$ of a triangle be $2 \hat{i}+2 \hat{j}+\hat{k}$,$\hat{i}+2 \hat{j}+2 \hat{k}$ and $2 \hat{i}+\hat{j}+2 \hat{k}$ respectively. Let $l_1, l_2$ and $l_3$ be the lengths of perpendiculars drawn from the orthocenter of the triangle on the sides $AB, BC$ and $CA$ respectively,then $l_1^2+l_2^2+l_3^2$ equals:

If the position vectors of the points $A, B, C, D$ are $2i + 3j + 5k, i + 2j + 3k, -5i + 4j - 2k$ and $i + 10j + 10k$ respectively,then:

Let $ABCDEF$ be a regular hexagon with the vertices $A, B, C, D, E, F$ in counter-clockwise order. If $O$ is the center of $ABCDEF$,then the vector $\vec{AO}$ is equal to which of the following?