The Cartesian equation of the line passing through the points (1, −2, 3) and (4, 0, −1) is
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → \(\displaystyle\frac{x-1}{3}=\frac{y+2}{2}=\frac{z-3}{-4}\)
The direction ratios of the line through two points are the differences of their coordinates:
\(4-1 = 3,\quad 0-(-2) = 2,\quad -1-3 = -4\)
Using the point (1, −2, 3):
\( \displaystyle \frac{x-1}{3}=\frac{y+2}{2}=\frac{z-3}{-4} \)
Why the other options are incorrect
Option (2) \(\displaystyle\frac{x-1}{5}=\frac{y+2}{-2}=\frac{z-3}{2}\) — Incorrect. The denominators 5, −2, 2 are the sums of the coordinates, not the differences.
Option (3) \(\displaystyle\frac{x+1}{3}=\frac{y-2}{2}=\frac{z+3}{-4}\) — Incorrect. This line passes through (−1, 2, −3); the point (1, −2, 3) gives \(\frac{2}{3} \ne \frac{-4}{2}\), so it is not on it.
Option (4) \(\displaystyle\frac{x-4}{3}=\frac{y}{2}=\frac{z+1}{4}\) — Incorrect. The z-direction ratio is −4, not 4.