The altitude through vertex $A$ of $\triangle A B C$ with position vectors of points $A, B, C$ as $\bar{a}, \bar{b}, \bar{c}$ respectively is

$$ \int \frac{\mathrm{d} x}{(x+\mathrm{a})^{\frac{9}{7}}(x-\mathrm{b})^{\frac{5}{7}}}= $$

The lines $\bar{r}=(\hat{i}+\hat{j}-\hat{k})+\lambda(3 \hat{i}-\hat{j})$ and $\overline{\mathrm{r}}=(4 \hat{\mathrm{i}}-\hat{\mathrm{k}})+\mu(2 \hat{\mathrm{i}}+3 \hat{\mathrm{k}})$ are

If $\overline{\mathrm{b}}$ and $\overline{\mathrm{c}}$ are unit vectors and $|\overline{\mathrm{a}}|=7$, $\overline{\mathrm{a}} \times(\overline{\mathrm{b}} \times \overline{\mathrm{c}})+\overline{\mathrm{b}} \times(\overline{\mathrm{c}} \times \overline{\mathrm{a}})=\frac{1}{2} \overline{\mathrm{a}}$, then angle between the vectors $\bar{a}$ and $\bar{c}$ and angle between the vectors $\overline{\mathrm{b}}$ and $\overline{\mathrm{c}}$ are respectively