1. **State the problem:** We are given two vector equations of lines in 3D: $$\vec{r}_1 = 3 \hat{i} + \hat{j} - 2 \hat{k} + \lambda (\hat{i} - \hat{j} - 2 \hat{k})$$ $$\vec{r}_2 = 2 \hat{i} - \hat{j} - 56 \hat{k} + \mu (3 \hat{i} - 5 \hat{j} - 4 \hat{k})$$ We want to analyze these lines, for example, to find if they intersect, are parallel, or skew. 2. **Rewrite the parametric form:** For line 1: $$x = 3 + \lambda, \quad y = 1 - \lambda, \quad z = -2 - 2\lambda$$ For line 2: $$x = 2 + 3\mu, \quad y = -1 - 5\mu, \quad z = -56 - 4\mu$$ 3. **Set the coordinates equal to find intersection:** We want to find $\lambda$ and $\mu$ such that: $$3 + \lambda = 2 + 3\mu$$ $$1 - \lambda = -1 - 5\mu$$ $$-2 - 2\lambda = -56 - 4\mu$$ 4. **Solve the system:** From the first equation: $$\lambda - 3\mu = -1$$ From the second: $$-\lambda + 5\mu = -2$$ Add the two equations: $$\cancel{\lambda} - 3\mu - \cancel{\lambda} + 5\mu = -1 - 2$$ $$2\mu = -3 \implies \mu = -\frac{3}{2}$$ 5. **Find $\lambda$ using $\mu$:** From first equation: $$\lambda - 3\left(-\frac{3}{2}\right) = -1$$ $$\lambda + \frac{9}{2} = -1$$ $$\lambda = -1 - \frac{9}{2} = -\frac{11}{2}$$ 6. **Check the third equation for consistency:** $$-2 - 2\left(-\frac{11}{2}\right) = -56 - 4\left(-\frac{3}{2}\right)$$ $$-2 + 11 = -56 + 6$$ $$9 \neq -50$$ 7. **Conclusion:** Since the third equation is not satisfied, the lines do not intersect. 8. **Check if lines are parallel:** Direction vectors: $$\vec{d}_1 = (1, -1, -2), \quad \vec{d}_2 = (3, -5, -4)$$ Check if $\vec{d}_2 = k \vec{d}_1$ for some scalar $k$: $$\frac{3}{1} = 3, \quad \frac{-5}{-1} = 5, \quad \frac{-4}{-2} = 2$$ Since these ratios are not equal, lines are not parallel. 9. **Final answer:** The lines are skew (neither intersecting nor parallel). **Answer:** The lines do not intersect and are skew lines in 3D space.