CBSE 2025, Set 3

Question

**Case Study - 2**
An engineer is designing a new metro rail network in a city.

![Diagram Metro Rail](https://www.edvaya.com/images/cbse/2025/mathematics/set3/q37-metro-rail.webp)

Initially, two metro lines, Line A and Line B, each consisting of multiple stations are designed. The track for Line A is represented by
$l_1: \frac{x-2}{3} = \frac{y+1}{-2} = \frac{z-3}{4}$ while the track for Line B is represented by
$l_2: \frac{x-1}{2} = \frac{y-3}{1} = \frac{z+2}{-3}$.

Based on the above information, answer the following question:
(i) Find whether the two metro tracks are parallel.

Edvaya · Accepted Answer

The correct answer is Subjective. **Step 1:** Extract the direction vectors of the two lines. Let the direction vector of Line A ($l_1$) be $\vec{b_1}$ and the direction vector of Line B ($l_2$) be $\vec{b_2}$. From the given cartesian equations, the denominators represent the direction ratios: For Line A ($l_1$): $\vec{b_1} = 3\hat{i} - 2\hat{j} + 4\hat{k}$ For Line B ($l_2$): $\vec{b_2} = 2\hat{i} + 1\hat{j} - 3\hat{k}$ **Step 2:** Check for proportionality. Two lines are parallel if and only if their direction vectors are proportional, i.e., $$\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}$$ Substitute the direction ratios of $l_1$ and $l_2$: $$\frac{3}{2} 
e \frac{-2}{1} 
e \frac{4}{-3}$$ Since the direction ratios are not proportional, the two metro tracks are **not parallel**.

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