Let $L_1$ and $L_2$ be two lines, represented as,

$L_1: \vec{r} = \hat{i} + \hat{j} + \lambda(2\hat{i} - \hat{j} + \hat{k})$ and $L_2: \vec{r} = 2\hat{i} + \hat{j} - \hat{k} + \mu(4\hat{i} - 2\hat{j} + 2\hat{k})$, where $\lambda$ and $\mu$ 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:

Two lines in 3D space are given in vector form. A line equation $\vec{r} = \vec{a} + \lambda\vec{b}$ has position vector $\vec{a}$ (a point on the line) and direction vector $\vec{b}$.

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For Line $L_1$: $\vec{r} = \hat{i} + \hat{j} + \lambda(2\hat{i} - \hat{j} + \hat{k})$

Point on $L_1$ (when $\lambda = 0$): $(1, 1, 0)$

Direction vector of $L_1$: $\vec{d_1} = (2, -1, 1)$

For Line $L_2$: $\vec{r} = 2\hat{i} + \hat{j} - \hat{k} + \mu(4\hat{i} - 2\hat{j} + 2\hat{k})$

Point on $L_2$ (when $\mu = 0$): $(2, 1, -1)$

Direction vector of $L_2$: $\vec{d_2} = (4, -2, 2)$

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Checking Statement (C): Does $L_1$ pass through $(1, 1, 0)$?

When $\lambda = 0$, the point is $(1, 1, 0)$.

Statement (C) is TRUE.

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Checking Statement (D): Does $L_2$ pass through $(2, 1, -1)$?

When $\mu = 0$, the point is $(2, 1, -1)$.

Statement (D) is TRUE.

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Checking Statement (B): Are $L_1$ and $L_2$ parallel?

Two lines are parallel if their direction vectors are scalar multiples of each other.

Comparing $\vec{d_1} = (2, -1, 1)$ and $\vec{d_2} = (4, -2, 2)$:

$\vec{d_2} = 2\vec{d_1}$

$(4, -2, 2) = 2(2, -1, 1)$

The lines have the same direction.

Statement (B) is TRUE.

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Checking Statement (A): Are $L_1$ and $L_2$ perpendicular?

Two lines are perpendicular if the dot product of their direction vectors equals zero.

$\vec{d_1} \cdot \vec{d_2} = (2)(4) + (-1)(-2) + (1)(2)$

$= 8 + 2 + 2$

$= 12$

Since $12 \neq 0$, the lines are not perpendicular.

Statement (A) is FALSE.

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Statements (B), (C), and (D) are TRUE.