Consider the lines
\(L_1: \displaystyle\frac{x-1}{1}=\frac{y+2}{-2}=\frac{z-3}{2}\), \(L_2: \vec{r} = (\hat{\imath}-\hat{k})+\lambda(2\hat{\imath}+\hat{\jmath})\),
\(L_3: \displaystyle\frac{x+1}{-2}=\frac{y-4}{4}=\frac{z+1}{-4}\), \(L_4: \displaystyle\frac{x}{2}=\frac{y}{2}=\frac{z-5}{1}\)
(A) \(L_1\) is parallel to \(L_3\)
(B) \(L_1\) is perpendicular to \(L_2\)
(C) \(L_2\) is perpendicular to \(L_4\)
(D) \(L_3\) is perpendicular to \(L_4\)
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → (A), (B) and (D) only
The direction ratios of the four lines are
\(L_1: 1, -2, 2\qquad L_2: 2, 1, 0\)
\(L_3: -2, 4, -4\qquad L_4: 2, 2, 1\)
Two lines are parallel when their direction ratios are proportional, and perpendicular when \(a_1a_2+b_1b_2+c_1c_2 = 0\).
\(L_3\): \(-2, 4, -4 = -2 \times (1, -2, 2)\)
\(L_1, L_2\): \((1)(2)+(-2)(1)+(2)(0) = 0\)
\(L_2, L_4\): \((2)(2)+(1)(2)+(0)(1) = 6\)
\(L_3, L_4\): \((-2)(2)+(4)(2)+(-4)(1) = 0\)
So the direction ratios of \(L_3\) are −2 times those of \(L_1\), and the dot sums are 0 for \(L_1, L_2\), 6 for \(L_2, L_4\) and 0 for \(L_3, L_4\).
Checking the statements
(A) \(L_1\) is parallel to \(L_3\) — Correct. −2, 4, −4 is −2 times 1, −2, 2.
(B) \(L_1\) is perpendicular to \(L_2\) — Correct. \(2-2+0 = 0\).
(C) \(L_2\) is perpendicular to \(L_4\) — Incorrect. \(4+2+0 = 6 \ne 0\).
(D) \(L_3\) is perpendicular to \(L_4\) — Correct. \(-4+8-4 = 0\).