The equation of line passing through the points (2, -1, 3) and (4, 2, 1) is
(A) \( \displaystyle\frac{x-2}{2}=\frac{y+1}{3}=\frac{z-3}{-2} \)
(B) \( \displaystyle\frac{x-4}{3}=\frac{y-2}{-2}=\frac{z-1}{2} \)
(C) \( \vec{r}=\left(2\hat{\imath}-\hat{\jmath}+3\hat{k}\right)+\lambda\left(2\hat{\imath}+3\hat{\jmath}-2\hat{k}\right) \)
(D) \( \displaystyle\frac{x-4}{-2}=\frac{y-2}{-3}=\frac{z-1}{2} \)
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → (A), (C) and (D) only
The direction ratios of the line are the differences of the coordinates:
\( \displaystyle (4-2,\ 2-(-1),\ 1-3)=(2,\ 3,\ -2) \)
The same line can be written through either point, with direction ratios (2, 3, −2) or any non-zero multiple of them, including a negative one.
So a correct equation must pass through (2, −1, 3) or (4, 2, 1) and have direction ratios proportional to (2, 3, −2).
Checking the statements
(A) \( \frac{x-2}{2}=\frac{y+1}{3}=\frac{z-3}{-2} \) — Correct. Point (2, −1, 3), direction ratios (2, 3, −2).
(B) \( \frac{x-4}{3}=\frac{y-2}{-2}=\frac{z-1}{2} \) — Incorrect. (3, −2, 2) is not proportional to (2, 3, −2).
(C) \( \vec{r}=\left(2\hat{\imath}-\hat{\jmath}+3\hat{k}\right)+\lambda\left(2\hat{\imath}+3\hat{\jmath}-2\hat{k}\right) \) — Correct. Vector form with point (2, −1, 3) and direction (2, 3, −2).
(D) \( \frac{x-4}{-2}=\frac{y-2}{-3}=\frac{z-1}{2} \) — Correct. Point (4, 2, 1), and (−2, −3, 2) = −1 × (2, 3, −2).