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:
(iii) (a) To connect the stations, a pedestrian pathway perpendicular to the two metro lines is to be constructed which passes through point $(3, 2, 1)$. Determine the equation of the pedestrian walkway.

Edvaya · Accepted Answer

The correct answer is Subjective. **Step 1:** Find the direction vector of the pedestrian walkway. Let the direction vector of the pedestrian walkway be $\vec{b}$. Since the walkway is strictly perpendicular to both Line A ($l_1$) and Line B ($l_2$), its direction vector must be perpendicular to the direction vectors of both lines, $\vec{b_1}$ and $\vec{b_2}$. Therefore, $\vec{b}$ is parallel to the cross product $\vec{b_1} 	imes \vec{b_2}$. From the given equations: $\vec{b_1} = 3\hat{i} - 2\hat{j} + 4\hat{k}$ $\vec{b_2} = 2\hat{i} + 1\hat{j} - 3\hat{k}$ Calculate the cross product: $$\vec{b} = \vec{b_1} 	imes \vec{b_2} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \ 3 & -2 & 4 \ 2 & 1 & -3 \end{vmatrix}$$ $$= \hat{i}((-2)(-3) - (4)(1)) - \hat{j}((3)(-3) - (4)(2)) + \hat{k}((3)(1) - (-2)(2))$$ $$= \hat{i}(6 - 4) - \hat{j}(-9 - 8) + \hat{k}(3 - (-4))$$ $$= 2\hat{i} + 17\hat{j} + 7\hat{k}$$ So, the direction ratios of the walkway are $(2, 17, 7)$. **Step 2:** Formulate the equation of the walkway. The walkway passes through the given point $(x_1, y_1, z_1) = (3, 2, 1)$. Using the standard cartesian equation formula $\frac{x - x_1}{a} = \frac{y - y_1}{b} = \frac{z - z_1}{c}$: $$\frac{x - 3}{2} = \frac{y - 2}{17} = \frac{z - 1}{7}$$ This is the required equation of the pedestrian walkway.

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