The vector equation of the line passing through the origin and the point (3, −1, 2) is
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → \(\vec{r} = \lambda(3\hat{\imath}-\hat{\jmath}+2\hat{k})\)
A line through the points with position vectors \(\vec{a}\) and \(\vec{b}\) is \(\vec{r} = \vec{a}+\lambda(\vec{b}-\vec{a})\).
Here the first point is the origin, so \(\vec{a} = \vec{0}\), and the second has position vector \(3\hat{\imath}-\hat{\jmath}+2\hat{k}\):
\(\vec{r} = \vec{0}+\lambda(3\hat{\imath}-\hat{\jmath}+2\hat{k})\)
\(\vec{r} = \lambda(3\hat{\imath}-\hat{\jmath}+2\hat{k})\)
\(\lambda = 0\) gives the origin and \(\lambda = 1\) gives (3, −1, 2).
Why the other options are incorrect
Option (1) \(\vec{r} = (3\hat{\imath}-\hat{\jmath}+2\hat{k})+\lambda\hat{k}\) — Incorrect. Every point of this line has x = 3, so it never reaches the origin.
Option (3) \(\vec{r} = \lambda(3\hat{\imath}+\hat{\jmath}+2\hat{k})\) — Incorrect. The y-component has the wrong sign; this line passes through (3, 1, 2), not (3, −1, 2).
Option (4) \(\vec{r} = \hat{\imath}+\lambda(3\hat{\imath}-\hat{\jmath}+2\hat{k})\) — Incorrect. This passes through (1, 0, 0) and is parallel to the correct line, so it misses the origin.