If the equation of the line passing through the point (0,-1/2, 0) and perpendicular to the lines r =λ( i + a j + b k ) and r =( i - j -6 k )+μ(- b i…

If the equation of the line passing through the point $\displaystyle \left(0,-\frac{1}{2}, 0\right)$ and perpendicular to the lines $\displaystyle \vec{r}=\lambda(\hat{i}+\mathrm{a} \hat{j}+\mathrm{b} \hat{k})$ and $\displaystyle \vec{r}=(\hat{i}-\hat{j}-6 \hat{k})+\mu(-\mathrm{b} \hat{i}+\mathrm{a} \hat{j}+5 \hat{k})$ is $\displaystyle \frac{x-1}{-2}=\frac{y+4}{\mathrm{~d}}=\frac{z-\mathrm{c}}{-4}$, then $\displaystyle \mathrm{a}+\mathrm{b}+\mathrm{c}+\mathrm{d}$ is equal to :