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(D) The diagonal $OP$ passes through $(0, 0, 0)$ and $(3, 4, 5)$. Its direction vector is $\vec{b}_1 = 3\hat{i} + 4\hat{j} + 5\hat{k}$. The equation o

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$\frac{12}{5}$

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(D) The diagonal $OP$ passes through $(0, 0, 0)$ and $(3, 4, 5)$. Its direction vector is $\vec{b}_1 = 3\hat{i} + 4\hat{j} + 5\hat{k}$. The equation of $OP$ is $\frac{x}{3} = \frac{y}{4} = \frac{z}{5}$.An edge parallel to the $z$-axis not passing through $O(0, 0, 0)$ or $P(3, 4, 5)$ must pass through the vertex $(3, 0, 0)$ or $(0, 4, 0)$. Let us consider the edge passing through $(3, 0, 0)$. Its direction vector is $\vec{b}_2 = \hat{k} = (0, 0, 1)$.The shortest distance $d$ between two skew lines $\vec{r} = \vec{a}_1 + t\vec{b}_1$ and $\vec{r} = \vec{a}_2 + s\vec{b}_2$ is given by $d = \frac{|(\vec{a}_2 - \vec{a}_1) \cdot (\vec{b}_1 	imes \vec{b}_2)|}{|\vec{b}_1 	imes \vec{b}_2|}$.Here,$\vec{a}_1 = (0, 0, 0)$,$\vec{a}_2 = (3, 0, 0)$,$\vec{b}_1 = (3, 4, 5)$,and $\vec{b}_2 = (0, 0, 1)$.$\vec{b}_1 	imes \vec{b}_2 = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \ 3 & 4 & 5 \ 0 & 0 & 1 \end{vmatrix} = 4\hat{i} - 3\hat{j}$.$|\vec{b}_1 	imes \vec{b}_2| = \sqrt{4^2 + (-3)^2} = \sqrt{16 + 9} = 5$.$(\vec{a}_2 - \vec{a}_1) \cdot (\vec{b}_1 	imes \vec{b}_2) = (3, 0, 0) \cdot (4, -3, 0) = 12$.Therefore,$d = \frac{|12|}{5} = \frac{12}{5}$.

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