Appeared in: 2024 (6 sets)
The Cartesian equation of the line passing through the point $(1, -3, 2)$ and parallel to the line: $\vec{r} = $2 + \lambda$\hat{i} + \lambda\hat{j} + $2\lambda - 1$\hat{k}$ is (A) $\frac{x - 1}{2} = \frac{y + 3}{0} = \frac{z - 2}{-1}$ (B) $\frac{x + 1}{1} = \frac{y - 3}{1} = \frac{z + 2}{2}$ (C) $\frac{x + 1}{2} = \frac{y - 3}{0} = \frac{z + 2}{-1}$ (D) $\frac{x - 1}{1} = \frac{y + 3}{1} = \frac{z - 2}{2}$
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Official Evaluation Steps
Step 1 (Formula / Governing Law):
The Cartesian equation of a line passing through a point $(x_1, y_1, z_1)$ and parallel to a vector $\vec{b} = a\hat{i} + b\hat{j} + c\hat{k}$ is given by:
$$\frac{x - x_1}{a} = \frac{y - y_1}{b} = \frac{z - z_1}{c}$$(0.25 Mark)
Step 2 (Step-by-Step Working):
- Identify the point: The line passes through $(1, -3, 2)$. Thus, $x_1 = 1, y_1 = -3, z_1 = 2$.
- Identify the direction vector: The given line is $\vec{r} = (2 + \lambda)\hat{i} + (\lambda)\hat{j} + (2\lambda - 1)\hat{k}$. Rewriting this in standard form $\vec{r} = \vec{a} + \lambda\vec{b}$: $$\vec{r} = (2\hat{i} - \hat{k}) + \lambda(\hat{i} + \hat{j} + 2\hat{k})$$ The direction ratios of the parallel line are $(a, b, c) = (1, 1, 2)$.
- Substitute into the Cartesian formula: $$\frac{x - 1}{1} = \frac{y - (-3)}{1} = \frac{z - 2}{2}$$ $$\frac{x - 1}{1} = \frac{y + 3}{1} = \frac{z - 2}{2}$$
(0.5 Mark)
Step 3 (Final Answer):
Comparing the derived equation with the given options, it matches option (D).
Correct Option: (D)
$$\boxed{\frac{x - 1}{1} = \frac{y + 3}{1} = \frac{z - 2}{2}}$$(0.25 Mark)