The shortest distance between the lines whose vector equations are $\vec r=(\hat i+2\hat j+3\hat k)+\lambda(2\hat i+3\hat j+4\hat k)$ and $\vec r=(2\hat i+4\hat j+5\hat k)+\mu(4\hat i+6\hat j+8\hat k)$ is:

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

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The equation of plane which cuts equal intercepts of unit length on the coordinate axes is :

The direction cosines of the line which is perpendicular to the lines with direction ratios $1,-2,-2$ and $0,2,1$ are :

The distance of the plane $\vec{r} \cdot \left(\frac{2}{7}\hat{i} + \frac{3}{7}\hat{j} - \frac{6}{7}\hat{k}\right) = 2$ from the origin is

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.

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The distance between the two lines $L_1$ and $L_2$ given by $L_1: \dfrac{x-1}{2} = \dfrac{y-2}{3} = \dfrac{-z-4}{-6}$ and $L_2: \dfrac{x-3}{2} = \dfrac{y-3}{3} = \dfrac{z+5}{6}$ is