Step-by-Step Solution

1. Line 1: $2x = 3y = -z$. We can rewrite this as $\frac{x}{1/2} = \frac{y}{1/3} = \frac{z}{-1}$. Multiplying by 6 to clear fractions, we get $\frac{x}{3} = \frac{y}{2} = \frac{z}{-6}$. So the direction ratios of the first line are $a_1 = 3$, $b_1 = 2$, and $c_1 = -6$.
2. Line 2: $6x = -y = -4z$. We can rewrite this as $\frac{x}{1/6} = \frac{y}{-1} = \frac{z}{-1/4}$. Multiplying by 12 to clear fractions, we get $\frac{x}{2} = \frac{y}{-12} = \frac{z}{-3}$. So the direction ratios of the second line are $a_2 = 2$, $b_2 = -12$, and $c_2 = -3$.
3. Angle Formula: The angle $\theta$ between two lines with direction ratios $(a_1, b_1, c_1)$ and $(a_2, b_2, c_2)$ is given by: $$\cos \theta = \frac{|a_1a_2 + b_1b_2 + c_1c_2|}{\sqrt{a_1^2 + b_1^2 + c_1^2} \sqrt{a_2^2 + b_2^2 + c_2^2}}$$
4. Substitute: Plugging in the values, we get: $$\cos \theta = \frac{|(3)(2) + (2)(-12) + (-6)(-3)|}{\sqrt{3^2 + 2^2 + (-6)^2} \sqrt{2^2 + (-12)^2 + (-3)^2}}$$ $$\cos \theta = \frac{|6 - 24 + 18|}{\sqrt{9 + 4 + 36} \sqrt{4 + 144 + 9}}$$ $$\cos \theta = \frac{|0|}{\sqrt{49} \sqrt{157}}$$ $$\cos \theta = 0$$
5. Solve for $\theta$: Since $\cos \theta = 0$, we have $\theta = \frac{\pi}{2}$ or $90^\circ$.