Question

The Cartesian equation of the line $\vec{r}=(2\hat{i}-7\hat{j}+11\hat{k})+\lambda(3\hat{i}+7\hat{j}-13\hat{k})$ is:

• A. $\frac{x-2}{3}=\frac{y+7}{7}=\frac{z-11}{-13}$
• B. $\frac{x-2}{3}=\frac{y-7}{7}=\frac{z-11}{13}$
• C. $\frac{x+2}{3}=\frac{y-7}{7}=\frac{z+11}{-13}$
• D. $\frac{x+2}{3}=\frac{y+7}{7}=\frac{z-11}{-13}$
• E. $\frac{x+2}{3}=\frac{y}{13}=\frac{z-11}{-7}$

Hint:

Always ensure signs are reversed for the coordinates in the numerator $(x-x_1)$.

Answer 1

The Correct Option is A

Step 1: Concept
Vector line $\vec{r} = (x_1\hat{i} + y_1\hat{j} + z_1\hat{k}) + \lambda(a\hat{i} + b\hat{j} + c\hat{k})$ converts to $\frac{x-x_1}{a} = \frac{y-y_1}{b} = \frac{z-z_1}{c}$.

Step 2: Analysis
Passing point $(x_1, y_1, z_1) = (2, -7, 11)$. Direction ratios $(a, b, c) = (3, 7, -13)$.

Step 3: Conclusion
Equation is $\frac{x-2}{3} = \frac{y-(-7)}{7} = \frac{z-11}{-13} \implies \frac{x-2}{3} = \frac{y+7}{7} = \frac{z-11}{-13}$. Final Answer: (A)