If two vertices of a triangle are $\hat{i} - \hat{j}$ and $\hat{j} + \hat{k}$,then the third vertex can be

If $\overline{a} = m \overline{b} + n \overline{c}$,where $\overline{a} = 4 \hat{i} + 13 \hat{j} - 18 \hat{k}$,$\overline{b} = \hat{i} - 2 \hat{j} + 3 \hat{k}$,and $\overline{c} = 2 \hat{i} + 3 \hat{j} - 4 \hat{k}$,then $m + n =$

If $a, b, c$ are the position vectors of the vertices $A, B, C$ of the triangle $ABC,$ then the centroid of $\Delta ABC$ is

The direction cosine of the vector $\vec{a} = 3\hat{i} + 4\hat{j} + 5\hat{k}$ in the direction of the positive $x$-axis is:

$P$ is a point on the side $BC$ of the $\Delta ABC$ and $Q$ is a point such that $\overrightarrow{PQ}$ is the resultant of $\overrightarrow{AP}, \overrightarrow{PB}, \overrightarrow{PC}$. Then $ABQC$ is a

Let $\bar{a}, \bar{b}$ and $\bar{c}$ be non-coplanar vectors. If $P, Q, R$ and $S$ are four points with position vectors $-\bar{a}+4\bar{b}-3\bar{c}$,$3\bar{a}+2\bar{b}-5\bar{c}$,$-3\bar{a}+8\bar{b}-5\bar{c}$ and $-3\bar{a}+2\bar{b}+\bar{c}$ respectively,then the ordered pair $(x, y)$ of real numbers such that $\overline{PQ} = x \cdot \overline{PR} + y \cdot \overline{PS}$ is