The Cartesian equation of the line \(\vec{r} = (2\hat{\imath}-3\hat{k})+\lambda(\hat{\imath}+4\hat{\jmath}-2\hat{k})\) is
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → \(\displaystyle\frac{x-2}{1}=\frac{y}{4}=\frac{z+3}{-2}\)
In \(\vec{r} = \vec{a}+\lambda\vec{b}\), the vector \(\vec{a}\) gives a point of the line and \(\vec{b}\) gives its direction ratios.
\(\vec{a} = 2\hat{\imath}+0\hat{\jmath}-3\hat{k}\), so the point is (2, 0, −3)
\(\vec{b} = \hat{\imath}+4\hat{\jmath}-2\hat{k}\), so the direction ratios are 1, 4, −2
The Cartesian form \(\frac{x-x_1}{a}=\frac{y-y_1}{b}=\frac{z-z_1}{c}\) gives
\( \displaystyle \frac{x-2}{1}=\frac{y-0}{4}=\frac{z+3}{-2} \)
Why the other options are incorrect
Option (1) \(\displaystyle\frac{x+2}{1}=\frac{y}{4}=\frac{z-3}{-2}\) — Incorrect. The signs of the point are reversed; this line passes through (−2, 0, 3), which is not on the given line.
Option (3) \(\displaystyle\frac{x-1}{2}=\frac{y-4}{0}=\frac{z+2}{-3}\) — Incorrect. The point and the direction have been swapped; its direction 2, 0, −3 is not proportional to 1, 4, −2.
Option (4) \(\displaystyle\frac{x-2}{1}=\frac{y}{4}=\frac{z+3}{2}\) — Incorrect. The last direction ratio should be −2, not 2.