Let L be the line passing through the points A(1, 2, −1) and B(3, −2, 3).
(A) The direction ratios of L are 2, −4, 4
(B) The point (5, −6, 7) lies on L
(C) \(\vec{r} = (3\hat{\imath}-2\hat{\jmath}+3\hat{k})+\lambda(\hat{\imath}-2\hat{\jmath}+2\hat{k})\) is a vector equation of L
(D) L is perpendicular to the line \(\displaystyle\frac{x-1}{2}=\frac{y}{1}=\frac{z+3}{1}\)
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → (A), (B) and (C) only
The direction ratios of AB are the differences of the coordinates:
\(3-1 = 2,\quad -2-2 = -4,\quad 3-(-1) = 4\)
These are proportional to 1, −2, 2. A general point of L, starting from A, is
\((1+t,\ 2-2t,\ -1+2t)\)
\(t = 4\): \((5, -6, 7)\)
For (D), the other line has direction ratios 2, 1, 1:
\((1)(2)+(-2)(1)+(2)(1) = 2\)
So L has direction ratios 2, −4, 4 (or 1, −2, 2), (5, −6, 7) is the point with t = 4, and the dot sum with 2, 1, 1 is 2.
Checking the statements
(A) The direction ratios of L are 2, −4, 4 — Correct. These are B minus A.
(B) The point (5, −6, 7) lies on L — Correct. It is the point \(t = 4\).
(C) \(\vec{r} = (3\hat{\imath}-2\hat{\jmath}+3\hat{k})+\lambda(\hat{\imath}-2\hat{\jmath}+2\hat{k})\) is a vector equation of L — Correct. It passes through B and its direction 1, −2, 2 is that of L.
(D) L is perpendicular to \(\frac{x-1}{2}=\frac{y}{1}=\frac{z+3}{1}\) — Incorrect. \(2-2+2 = 2 \ne 0\).