The vector equation of the line passing through the point A(1, −2, 3) and parallel to the line joining the points B(2, 0, −1) and C(4, −3, 1) is
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → \(\vec{r} = (\hat{\imath}-2\hat{\jmath}+3\hat{k})+\lambda(2\hat{\imath}-3\hat{\jmath}+2\hat{k})\)
A line parallel to BC has the direction of \(\overrightarrow{BC}\):
\(\overrightarrow{BC} = (4-2)\hat{\imath}+(-3-0)\hat{\jmath}+(1-(-1))\hat{k}\)
\(\overrightarrow{BC} = 2\hat{\imath}-3\hat{\jmath}+2\hat{k}\)
The line passes through A, whose position vector is \(\hat{\imath}-2\hat{\jmath}+3\hat{k}\). In the form \(\vec{r} = \vec{a}+\lambda\vec{b}\):
\(\vec{r} = (\hat{\imath}-2\hat{\jmath}+3\hat{k})+\lambda(2\hat{\imath}-3\hat{\jmath}+2\hat{k})\)
Why the other options are incorrect
Option (2) \(\vec{r} = (2\hat{\imath}-3\hat{\jmath}+2\hat{k})+\lambda(\hat{\imath}-2\hat{\jmath}+3\hat{k})\) — Incorrect. The point and the direction have been swapped.
Option (3) \(\vec{r} = (2\hat{\imath}-\hat{k})+\lambda(2\hat{\imath}-3\hat{\jmath}+2\hat{k})\) — Incorrect. This is the line BC itself; it does not pass through A.
Option (4) \(\vec{r} = (\hat{\imath}-2\hat{\jmath}+3\hat{k})+\lambda(6\hat{\imath}-3\hat{\jmath})\) — Incorrect. \(6\hat{\imath}-3\hat{\jmath}\) is \(\vec{b}+\vec{c}\), not \(\vec{c}-\vec{b}\).