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

The area of the parallelogram whose diagonals are represented by the vectors $$\bar{a}=3 \hat{i}-\hat{j}-2 \hat{k}$$ and $$\bar{b}=-\hat{i}+3 \hat{j}-3 \hat{k}$$ is

If $$\overline{\mathrm{a}}+\overline{\mathrm{b}}+\overline{\mathrm{c}}=\overline{0}$$ with $$|\overline{\mathrm{a}}|=3,|\overline{\mathrm{b}}|=5$$ and $$|\overline{\mathrm{c}}|=7$$, then angle between $$\overline{\mathrm{a}}$$ and $$\overline{\mathrm{b}}$$ is

If $$|\bar{u}|=2$$ and $$\bar{u}$$ makes angles of $$60^{\circ}$$ and $$120^{\circ}$$ with axes $$\mathrm{OX}$$ and $$\mathrm{OY}$$ in the origin, then $$\bar{u}=$$