Given are two lines:$L_1: 2x = 3y = -z,L_2: 6x = -y = -4z$. Find the angle between the two lines.

$90^\circ$

The correct answer is Option (3) → $90^\circ$ ##

Rewrites the equation of $L_1$ in cartesian form as:

$\frac{x}{3} = \frac{y}{2} = \frac{z}{-6}$

Rewrites the equation of $L_2$ in cartesian form as:

$\frac{x}{2} = \frac{y}{-12} = \frac{z}{-3}$

Identifies the direction cosines of both the lines as $(3,2,-6)$ and $(2,-12,-3)$.

Finds the cosine of the angle between the two lines as:

$\cos \theta = \left| \frac{6 - 24 + 18}{\sqrt{49 \sqrt{157}}} \right| = 0$

Concludes that the angle between the two lines is $90^\circ$.