Let $L_1$ and $L_2$ be two lines, represented as, $L_1:\vec r=\hat i+\hat j+λ(2\hat i-\hat j+\hat k)$ and $L_2:\vec r = 2\hat i + \hat j − \hat k + μ(4\hat i - 2\hat j + 2\hat k)$, where $λ$ and $μ$ are scalars. Then which of the following are true?
(A) $L_1$ is perpendicular to $L_2$
(B) $L_1$ is parallel to $L_2$
(C) $L_1$ passes through the point (1, 1, 0)
(D) $L_2$ passes through the point (2, 1, -1)
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → (B), (C) and (D) only
L1: $\vec{r} = \hat{i} + \hat{j} + \lambda(2\hat{i} - \hat{j} + \hat{k})$
L2: $\vec{r} = 2\hat{i} + \hat{j} - \hat{k} + \mu(4\hat{i} - 2\hat{j} + 2\hat{k})$
Direction vector of L1: $\vec{d}_1 = \langle 2, -1, 1 \rangle$
Direction vector of L2: $\vec{d}_2 = \langle 4, -2, 2 \rangle$
$\vec{d}_2 = 2 \cdot \vec{d}_1 \Rightarrow$ L1 and L2 are parallel
⟹ (B) is true
Check perpendicularity: $\vec{d}_1 \cdot \vec{d}_2 = 2 \cdot 4 + (-1) \cdot (-2) + 1 \cdot 2 = 8 + 2 + 2 = 12 \ne 0$
⟹ (A) is false
Check if L1 passes through (1, 1, 0): Point on L1 when $\lambda = 0$ is $(1, 1, 0)$
⟹ (C) is true
Check if L2 passes through (2, 1, -1): Point on L2 when $\mu = 0$ is $(2, 1, -1)$
⟹ (D) is true