Which of the following statements are true?

(A) The vector equation of the line through the point (5, 2, -4) and parallel to the vector $3\hat{i} + 2\hat{j} - 8\hat{k}$ is $\vec{r} = (5\hat{i} + 2\hat{j} - 4\hat{k}) + \lambda(3\hat{i} + 2\hat{j} - 8\hat{k})$

(B) Vector form of the equation of line $\frac{x-5}{3} = \frac{y+4}{7} = \frac{z-6}{2}$ is $\vec{r} = (5\hat{i} - 4\hat{j} + 6\hat{k}) + \lambda(3\hat{i} + 7\hat{j} + 2\hat{k})$

(C) The direction cosines of z-axis are (1, 1,0).

(D) If a line has direction ratios 2, -1, -2, then its direction cosines are -2/3, -1/3, -2/3.

Choose the correct answer from the options given below:

Checking statement (A):

The vector equation of a line through point $(x_0, y_0, z_0)$ parallel to vector $\vec{v}$ is:

$\vec{r} = \vec{r_0} + \lambda\vec{v}$

Given point: $(5, 2, -4)$, so $\vec{r_0} = 5\hat{i} + 2\hat{j} - 4\hat{k}$

Given direction vector: $\vec{v} = 3\hat{i} + 2\hat{j} - 8\hat{k}$

The equation becomes:

$\vec{r} = (5\hat{i} + 2\hat{j} - 4\hat{k}) + \lambda(3\hat{i} + 2\hat{j} - 8\hat{k})$

Statement (A) is TRUE.

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Checking statement (B):

The Cartesian form $\frac{x-5}{3} = \frac{y+4}{7} = \frac{z-6}{2}$ can be written as $\frac{x-5}{3} = \frac{y-(-4)}{7} = \frac{z-6}{2}$

Point on the line: $(5, -4, 6)$

Direction ratios: $(3, 7, 2)$

Vector form:

$\vec{r} = (5\hat{i} - 4\hat{j} + 6\hat{k}) + \lambda(3\hat{i} + 7\hat{j} + 2\hat{k})$

Statement (B) is TRUE.

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Checking statement (C):

The z-axis passes through the origin and is parallel to the vector $\hat{k} = 0\hat{i} + 0\hat{j} + 1\hat{k}$

Direction cosines are: $(0, 0, 1)$

Statement (C) claims direction cosines are $(1, 1, 0)$, which is FALSE.

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Checking statement (D):

Given direction ratios: $(2, -1, -2)$

Magnitude $= \sqrt{2^2 + (-1)^2 + (-2)^2}$

$= \sqrt{4 + 1 + 4}$

$= \sqrt{9}$

$= 3$

Direction cosines $= \left(\frac{2}{3}, \frac{-1}{3}, \frac{-2}{3}\right)$

Statement (D) claims direction cosines are $\left(\frac{-2}{3}, \frac{-1}{3}, \frac{-2}{3}\right)$, which is FALSE.

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Statements (A) and (B) are true.