If $$\overline{\mathrm{e}}_1, \overline{\mathrm{e}}_2$$ and $$\overline{\mathrm{e}}_1+\overline{\mathrm{e}}_2$$ are unit vectors, then the angle between $$\overline{\mathrm{e}}_1$$ and $$\overline{\mathrm{e}}_2$$ is
If $$y=\log \tan \left(\frac{x}{2}\right)+\sin ^{-1}(\cos x)$$, then $$\frac{d y}{d x}=$$
If $$\overline{\mathrm{a}}, \overline{\mathrm{b}} , \overline{\mathrm{c}}$$ are three vectors which are perpendicular to $$\overline{\mathrm{b}}+\overline{\mathrm{c}}, \overline{\mathrm{c}}+\overline{\mathrm{a}}$$ and $$\overline{\mathrm{a}}+\overline{\mathrm{b}}$$ respectively, such that $$|\bar{a}|=2,|\bar{b}|=3,|\bar{c}|=4$$, then $$|\bar{a}+\bar{b}+\bar{c}|=$$
If the lines $$\frac{x-1}{2}=\frac{y+1}{3}=\frac{z-1}{4}$$ and $$\frac{x-3}{1}=\frac{y-k}{2}=\frac{z}{1}$$ intersect, then the values of $$k$$ is