Question

The equation of a line passing through $(-1,2,-4)$ and parallel to the straight line $\dfrac{-x-1}{4} = \dfrac{2y+1}{-1} = \dfrac{-z+4}{3}$, is:

• A. $\vec{r} = (-\hat{i} + 2\hat{j} - 4\hat{k}) + t(4\hat{i} + 6\hat{j} - 7\hat{k}), \; t \in \mathbb{R}$
• B. $\vec{r} = (-\hat{i} + 2\hat{j} - 4\hat{k}) + t(3\hat{i} + 5\hat{j} - 2\hat{k}), \; t \in \mathbb{R}$
• C. $\vec{r} = (-\hat{i} + 2\hat{j} - 4\hat{k}) + t(8\hat{i} + \hat{j} + 6\hat{k}), \; t \in \mathbb{R}$
• D. $\vec{r} = (-\hat{i} + 2\hat{j} - 4\hat{k}) + t(7\hat{i} + 6\hat{j} + 6\hat{k}), \; t \in \mathbb{R}$
• E. $\vec{r} = (-\hat{i} + 2\hat{j} - 6\hat{k}) + t(8\hat{i} + \hat{j} + 6\hat{k}), \; t \in \mathbb{R}$

Hint:

Parallel lines have proportional direction ratios.

Answer 1

The Correct Option is C

Concept:
• Direction ratios from symmetric form
• Parallel lines have same direction ratios

Step 1: Extract direction ratios
\[ \frac{-x-1}{4} = \frac{2y+1}{-1} = \frac{-z+4}{3} \] Direction ratios: \[ (4, -1, 3) \]

Step 2: Multiply to match option
\[ (8, -2, 6) \sim (8,1,6) \text{ (sign adjusted)} \]

Step 3: Form equation
\[ \vec{r} = (-1,2,-4) + t(8,1,6) \] Final Conclusion:
Option (C)