A line passes through the point with position vector \(2\hat{\imath} - \hat{\jmath} + 4\hat{k}\) and is in the direction of the vector \(\hat{\imath} + \hat{\jmath} - 2\hat{k}\). The equation of the line in Cartesian form is:
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → \(\displaystyle\frac{x-2}{1} = \frac{y+1}{1} = \frac{4-z}{2}\)
The line passes through the point \((2, -1, 4)\) and has direction ratios 1, 1, −2. The Cartesian form of a line through \((x_1, y_1, z_1)\) with direction ratios a, b, c is \(\frac{x - x_1}{a} = \frac{y - y_1}{b} = \frac{z - z_1}{c}\):
\( \displaystyle \frac{x-2}{1} = \frac{y+1}{1} = \frac{z-4}{-2} \)
Multiplying the numerator and the denominator of the last fraction by −1:
\( \displaystyle \frac{z-4}{-2} = \frac{4-z}{2} \)
\( \displaystyle \frac{x-2}{1} = \frac{y+1}{1} = \frac{4-z}{2} \)
Why the other options are incorrect
Option (2) \(\displaystyle\frac{x+2}{1} = \frac{y-1}{1} = \frac{z-4}{2}\) — Incorrect. It passes through \((-2, 1, 4)\), not \((2, -1, 4)\), and its last direction ratio is 2, not −2.
Option (3) \(\displaystyle\frac{x-2}{1} = \frac{y-1}{1} = \frac{z-4}{2}\) — Incorrect. It passes through \((2, 1, 4)\), and its last direction ratio is 2, not −2.
Option (4) \(\displaystyle\frac{x-2}{1} = \frac{y+1}{1} = \frac{z-4}{2}\) — Incorrect. The point is right, but the sign of the last direction ratio is lost: it gives direction 1, 1, 2 instead of 1, 1, −2.