1. **Problem Statement:** Verify if the given information about lines L and M is correct based on the provided direction ratios, parametric equations, and symmetric equations. 2. **Line L:** - Given vector form: $$\mathbf{r} = (3\mathbf{i} + 2\mathbf{j} - \mathbf{k}) + t(6\mathbf{i} - 4\mathbf{j} - 3\mathbf{k})$$ - Direction ratios from vector form: $(6, -4, -3)$ - Provided direction ratios: $(6 - 3, -4 - 2, -3 + 1) = (3, -6, -2)$ which is incorrect because direction ratios should be the vector multiplying $t$, i.e., $(6, -4, -3)$, not differences. - Parametric equations given: $$x = 3 + 3t,\quad y = 2 - 6t,\quad z = -1 + 2t$$ which do not match the direction ratios $(6, -4, -3)$ but rather $(3, -6, 2)$. - Symmetric form given: $$\frac{x - 3}{3} = \frac{y - 2}{-6} = \frac{z + 1}{-2} = t$$ which matches the parametric equations but not the original vector form direction ratios. **Conclusion for L:** The original vector form direction ratios $(6, -4, -3)$ do not match the parametric and symmetric forms direction ratios $(3, -6, 2)$. So the information is inconsistent for line L. 3. **Line M:** - Given vector form: $$\mathbf{r} = (5\mathbf{i} + 4\mathbf{j} - 7\mathbf{k}) + s(14\mathbf{i} - 6\mathbf{j} + 2\mathbf{k})$$ - Direction ratios from vector form: $(14, -6, 2)$ - Provided direction ratios: $(14 - 5, -6 - 4, 2 + 7) = (9, -10, 9)$ which is incorrect as direction ratios should be the vector multiplying $s$, i.e., $(14, -6, 2)$. - Parametric equations given: $$x = 5 + 9s,\quad y = 4 - 10s,\quad z = -7 + 9s$$ which correspond to direction ratios $(9, -10, 9)$, not $(14, -6, 2)$. - Symmetric form given: $$\frac{x - 5}{9} = \frac{y - 4}{-10} = \frac{z + 7}{9} = s$$ which matches the parametric equations but not the original vector form direction ratios. **Conclusion for M:** The original vector form direction ratios $(14, -6, 2)$ do not match the parametric and symmetric forms direction ratios $(9, -10, 9)$. So the information is inconsistent for line M. 4. **Summary:** - The direction ratios in the vector forms for both lines L and M do not match the direction ratios implied by the parametric and symmetric equations. - The differences calculated (e.g., $14 - 5$) are not relevant for direction ratios; direction ratios come directly from the vector multiplied by the parameter. **Final answer:** The information provided is not all correct; there are inconsistencies between the vector forms and the parametric/symmetric forms for both lines L and M.