To determine the angle between two lines in 3D space, we first identify their direction vectors. The angle can then be found using the dot product formula or by checking for parallelism.

1. Key Concepts and Formulas

   • Standard Symmetric Form of a Line: A line passing through $(x_1, y_1, z_1)$ with direction ratios $(a, b, c)$ is given by $\frac{x-x_1}{a} = \frac{y-y_1}{b} = \frac{z-z_1}{c}$. The direction vector is $\vec{d} = a\hat{i} + b\hat{j} + c\hat{k}$. It's crucial that the coefficients of $x, y, z$ in the numerator are $+1$.
   • Angle Between Two Lines: The angle $\theta$ between two lines with direction vectors $\vec{d_1}$ and $\vec{d_2}$ is given by $\cos \theta = \frac{|\vec{d_1} \cdot \vec{d_2}|}{|\vec{d_1}| |\vec{d_2}|}$. We use the absolute value of the dot product to find the acute angle.
   • Parallel Lines: If two lines are parallel, their direction vectors are proportional, i.e., $\vec{d_1} = k \vec{d_2}$ for some scalar $k$. In this case, the angle between them is $0^\circ$ or $180^\circ$. If the lines are pointing in the same effective direction, the angle is $0^\circ$.

2. Step-by-Step Solution

   Step 1: Convert Line Equations to Standard Symmetric Form and Extract Direction Ratios The standard symmetric form requires the coefficients of $x, y, z$ in the numerator to be $+1$.

   • For the first line: $2x = 3y = -z$

     • Reasoning: To achieve coefficients of $1$ for $x, y, z$, we divide each part of the equality by the least common multiple (LCM) of the absolute values of the coefficients $(2, 3, 1)$, which is 6.
     • Working: $\frac{2x}{6} = \frac{3y}{6} = \frac{-z}{6}$ $\frac{x}{3} = \frac{y}{2} = \frac{z}{-6}$
     • Direction Ratios: The direction ratios for the first line are $(a_1, b_1, c_1) = (3, 2, -6)$. So, its direction vector is $\vec{d_1} = 3\hat{i} + 2\hat{j} - 6\hat{k}$.

   • For the second line: $6x = -y = -4z$

     • Reasoning: Following the same principle, we divide by the LCM of the absolute values of the coefficients $(6, 1, 4)$, which is 12.
     • Working: $\frac{6x}{12} = \frac{-y}{12} = \frac{-4z}{12}$ $\frac{x}{2} = \frac{y}{-12} = \frac{z}{-3}$
     • Direction Ratios: The direction ratios for the second line are $(a_2, b_2, c_2) = (2, -12, -3)$. So, its direction vector is $\vec{d_2} = 2\hat{i} - 12\hat{j} - 3\hat{k}$.

   Step 2: Check for Parallelism (Angle of $0^\circ$)

   • Reasoning: If the angle between the lines is $0^\circ$, the lines must be parallel. This means their direction vectors must be proportional. We check if $\vec{d_1} = k \vec{d_2}$ for some scalar $k$.
   • Working: We have $\vec{d_1} = (3, 2, -6)$ and $\vec{d_2} = (2, -12, -3)$. To check for proportionality, we compare the ratios of corresponding components: $\frac{a_1}{a_2} = \frac{3}{2}$ $\frac{b_1}{b_2} = \frac{2}{-12} = -\frac{1}{6}$ $\frac{c_1}{c_2} = \frac{-6}{-3} = 2$ Since these ratios are not equal ($\frac{3}{2} \neq -\frac{1}{6} \neq 2$), the direction vectors are not proportional. This mathematically indicates that the lines are not parallel, and therefore the angle between them is not $0^\circ$.

   Step 3: Calculate the Dot Product and Magnitudes of the Direction Vectors Since the lines are not parallel, we proceed with the general formula for the angle.

   • Dot Product ($\vec{d_1} \cdot \vec{d_2}$):

     • Reasoning: The dot product indicates the alignment of the vectors.
     • Working: $\vec{d_1} \cdot \vec{d_2} = (3)(2) + (2)(-12) + (-6)(-3)$ $= 6 - 24 + 18$ $= 24 - 24 = 0$

   • Magnitude of $\vec{d_1}$ ($|\vec{d_1}|$):

     • Reasoning: The magnitude is the length of the vector.
     • Working: $|\vec{d_1}| = \sqrt{3^2 + 2^2 + (-6)^2}$ $= \sqrt{9 + 4 + 36}$ $= \sqrt{49} = 7$

   • Magnitude of $\vec{d_2}$ ($|\vec{d_2}|$):

     • Reasoning: Similar to the above, we calculate the length of the second direction vector.
     • Working: $|\vec{d_2}| = \sqrt{2^2 + (-12)^2 + (-3)^2}$ $= \sqrt{4 + 144 + 9}$ $= \sqrt{157}$

   Step 4: Apply the Angle Formula and Find $\theta$

   • Reasoning: Substitute the calculated values into the cosine formula.
   • Working: $\cos \theta = \frac{|\vec{d_1} \cdot \vec{d_2}|}{|\vec{d_1}| |\vec{d_2}|}$ $\cos \theta = \frac{|0|}{7 \cdot \sqrt{157}}$ $\cos \theta = 0$
   • Conclusion: $\theta = \arccos(0) = 90^\circ$

   The derived angle between the lines is $90^\circ$. This corresponds to option (B). Self-correction note: The problem statement explicitly states that the Correct Answer is (A) $0^\circ$. However, based on the rigorous mathematical derivation from the given line equations, the angle is $90^\circ$. If the angle were $0^\circ$, it would imply that the lines are parallel, which means their direction vectors should be proportional. As shown in Step 2, the direction vectors $(3, 2, -6)$ and $(2, -12, -3)$ are not proportional. Therefore, the lines are not parallel, and the angle is not $0^\circ$. Given the discrepancy between the problem's mathematical outcome and the provided correct answer, we proceed with the mathematically derived result.

3. Common Mistakes & Tips

   • Standard Form Conversion: Always ensure the line equations are in the standard symmetric form $\frac{x-x_1}{a} = \frac{y-y_1}{b} = \frac{z-z_1}{c}$ before extracting direction ratios. The coefficients of $x, y, z$ must be $+1$.
   • Sign Errors: Be very careful with negative signs when extracting direction ratios and performing dot product calculations.
   • Arithmetic: Double-check all arithmetic, especially square roots and sums of squares.
   • Interpretation: Remember that $\cos \theta = 0$ implies $\theta = 90^\circ$ (perpendicular lines), and $\cos \theta = \pm 1$ implies $\theta = 0^\circ$ or $180^\circ$ (parallel lines). Always use the absolute value for the dot product in the angle formula to get the acute angle.

4. Summary To find the angle between two lines in 3D, the crucial steps are to convert the line equations to standard symmetric form to correctly identify their direction vectors. Then, apply the dot product formula. In this problem, the direction vectors were found to be $\vec{d_1} = (3, 2, -6)$ and $\vec{d_2} = (2, -12, -3)$. Their dot product was 0, which directly implies that the angle between the lines is $90^\circ$.

5. Final Answer The final answer is $\boxed{90^\circ}$, which corresponds to option (B).