The vector equation of the line passing through the point P(2, 3, 4) and meeting the z-axis at right angles is
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → \(\vec{r} = (2\hat{\imath}+3\hat{\jmath}+4\hat{k})+\lambda(2\hat{\imath}+3\hat{\jmath})\)
Let the line meet the z-axis at Q(0, 0, c). Then
\(\overrightarrow{QP} = 2\hat{\imath}+3\hat{\jmath}+(4-c)\hat{k}\)
It is perpendicular to the z-axis, direction \(\hat{k}\):
\(\overrightarrow{QP}\cdot\hat{k} = 4-c = 0\)
\(c = 4\)
So Q is (0, 0, 4), the direction is \(2\hat{\imath}+3\hat{\jmath}\), and the line through P is
\(\vec{r} = (2\hat{\imath}+3\hat{\jmath}+4\hat{k})+\lambda(2\hat{\imath}+3\hat{\jmath})\)
Why the other options are incorrect
Option (2) \(\vec{r} = (2\hat{\imath}+3\hat{\jmath}+4\hat{k})+\lambda(2\hat{\imath}+3\hat{\jmath}+4\hat{k})\) — Incorrect. This line meets the z-axis at the origin, but \((2\hat{\imath}+3\hat{\jmath}+4\hat{k})\cdot\hat{k} = 4 \ne 0\), so not at right angles.
Option (3) \(\vec{r} = (2\hat{\imath}+3\hat{\jmath}+4\hat{k})+\lambda\hat{k}\) — Incorrect. This line is parallel to the z-axis and never meets it.
Option (4) \(\vec{r} = 4\hat{k}+\lambda(3\hat{\imath}+2\hat{\jmath})\) — Incorrect. It meets the z-axis at right angles, but its x- and y-ratios are swapped, so it does not pass through P.