Consider a line $\vec{r} = (\hat{i} + 4\hat{j}) + \lambda(2\hat{i} - 2\hat{j} + 3\hat{k})$, then which of the following statements are correct?

(A) it passes through point (9, -4, 12)

(B) it passes through point (1, 4, -1)

(C) its direction cosine's are $\frac{2}{\sqrt{17}}, \frac{-2}{\sqrt{17}}, \frac{3}{\sqrt{17}}$

(D) its Cartesian equation is $\frac{x - 1}{2} = \frac{y - 4}{-2} = \frac{z}{3}$

Choose the correct answer from the options given below:

Given line: $\vec{r} = (\hat{i} + 4\hat{j}) + \lambda(2\hat{i} - 2\hat{j} + 3\hat{k})$

The line can be written as:

Point on line: $(1, 4, 0)$

Direction vector: $(2, -2, 3)$

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For point $(9, -4, 12)$:

$(x, y, z) = (1, 4, 0) + \lambda(2, -2, 3)$

$(x, y, z) = (1+2\lambda, 4-2\lambda, 3\lambda)$

From x-coordinate:

$1 + 2\lambda = 9$

$\lambda = 4$

From y-coordinate:

$4 - 2\lambda = -4$

$\lambda = 4$

From z-coordinate:

$3\lambda = 12$

$\lambda = 4$

All three coordinates give $\lambda = 4$.

Statement (A) is correct.

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For point $(1, 4, -1)$:

From x-coordinate:

$1 + 2\lambda = 1$

$\lambda = 0$

From y-coordinate:

$4 - 2\lambda = 4$

$\lambda = 0$

From z-coordinate:

$3\lambda = -1$

$\lambda = -\frac{1}{3}$

Different values of $\lambda$ are obtained.

Statement (B) is incorrect.

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Direction vector: $(2, -2, 3)$

Magnitude:

$|\vec{d}| = \sqrt{2^2 + (-2)^2 + 3^2}$

$|\vec{d}| = \sqrt{4 + 4 + 9}$

$|\vec{d}| = \sqrt{17}$

Direction cosines:

$\left(\frac{2}{\sqrt{17}}, \frac{-2}{\sqrt{17}}, \frac{3}{\sqrt{17}}\right)$

Statement (C) is correct.

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For a line through point $(x_0, y_0, z_0)$ with direction vector $(a, b, c)$:

$\frac{x - x_0}{a} = \frac{y - y_0}{b} = \frac{z - z_0}{c}$

Using point $(1, 4, 0)$ and direction $(2, -2, 3)$:

$\frac{x - 1}{2} = \frac{y - 4}{-2} = \frac{z - 0}{3}$

$\frac{x - 1}{2} = \frac{y - 4}{-2} = \frac{z}{3}$

Statement (D) is correct.

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Correct statements: (A), (C), and (D)