If the vector equation of the plane $$\bar{r}=(2 \hat{i}+\hat{k})+\lambda \hat{i}+\mu(\hat{i}+2 \hat{j}-3 \hat{k})$$ in scalar product form is given by $$\overline{\mathrm{r}} \cdot(3 \hat{\mathrm{j}}+2 \hat{\mathrm{k}})=\alpha$$ then $$\alpha=$$

If the lines $$\frac{x-1}{2}=\frac{y+1}{3}=\frac{z-1}{4}$$ and $$\frac{x-2}{1}=\frac{y+m}{2}=\frac{z-2}{1}$$ intersect each other, then value of m is

The length of perpendicular drawn from the point $$2 \hat{i}-\hat{j}+5 \hat{k}$$ to the line $$\overline{\mathrm{r}}=(11 \hat{i}-2 \hat{j}-8 \hat{k})+\lambda(10 \hat{i}-4 \hat{j}-11 \hat{k})$$ is

Equation of the plane passing through the point $$(1,2,3)$$ and parallel to the plane $$2 x+3 y-4 z=0 $$