The foot of the perpendicular drawn from the origin to a line L is the point F(2, −1, 2). Which of the following can be the line L?
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → \(\vec{r} = (2\hat{\imath}-\hat{\jmath}+2\hat{k})+\lambda(\hat{\imath}+4\hat{\jmath}+\hat{k})\)
F is the foot of the perpendicular from O to L exactly when F lies on L and OF is perpendicular to L. Here
\(\overrightarrow{OF} = 2\hat{\imath}-\hat{\jmath}+2\hat{k}\)
Option (2) passes through F (\(\lambda = 0\)), and its direction gives
\((2)(1)+(-1)(4)+(2)(1) = 0\)
so OF is perpendicular to the line.
Why the other options are incorrect
Option (1) \(\vec{r} = (2\hat{\imath}-\hat{\jmath}+2\hat{k})+\lambda(2\hat{\imath}-\hat{\jmath}+2\hat{k})\) — Incorrect. This line runs along OF and passes through the origin itself (\(\lambda = -1\)), so the foot from O is O, not F.
Option (3) \(\vec{r} = (\hat{\imath}+4\hat{\jmath}+\hat{k})+\lambda(2\hat{\imath}-\hat{\jmath}+2\hat{k})\) — Incorrect. F is not on it: \(\overrightarrow{OF}-(\hat{\imath}+4\hat{\jmath}+\hat{k}) =\hat{\imath}-5\hat{\jmath}+\hat{k}\) is not a multiple of \(2\hat{\imath}-\hat{\jmath}+2\hat{k}\).
Option (4) \(\vec{r} = (2\hat{\imath}-\hat{\jmath}+2\hat{k})+\lambda(\hat{\imath}+\hat{\jmath}+\hat{k})\) — Incorrect. It passes through F, but \(2-1+2 = 3 \ne 0\), so OF is not perpendicular to it.