$$O A B C$$ is a tetrahedron. If $$D, E$$ are the mid-points of $$O A$$ and $$B C$$ respectively, then $$\mathbf{D E}=$$

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

If $$P$$ and $$Q$$ are two points on the curve $$y=2^{x+2}$$ in the rectangular cartesian coordinate system such that $$\mathbf{O P} \cdot \hat{i}=-1, \mathrm{OQ} \cdot \hat{i}=2$$, then $$\mathrm{OQ}-4 \mathrm{OP}=$$

If the equation of the plane passing through the point $$A(-2,1,3)$$ and perpendicular to the vector $$3 \hat{i}+\hat{j}+5 \hat{k}$$ is $$a x+b y+c z+d=0$$, then $$\frac{a+b}{c+d}=$$