The image provided is blurry, making it difficult to read the exact coefficients for the vector equations. The question asks to "Show that $L_1$ and $L_2$ intersect", which implies they must intersect. However, with the most plausible reading of the equations, the lines do not intersect. To proceed with the problem as stated ("Show that $L_1$ and $L_2$ intersect"), I must assume there is a typo in the question's given equations. Based on common problem structures, a likely typo would be in the constant vector of $L_2$. Assumption: I will assume the constant vector for $L_2$ is $(2\mathbf{i} + 2\mathbf{j} + 3\mathbf{k})$ instead