If \( \mathrm{A}(\overline{\mathrm{a}}) ; \mathrm{B}(\overline{\mathrm{b}}) ; \mathrm{C}(\overline{\mathrm{c}}) \) and \( \mathrm{D}(\overline{\mathrm{d}}) \) are four points such that \( \bar{a}=-2 \hat{i}+4 \hat{j}+3 \hat{k} ; \quad \bar{b}=2 \hat{i}-8 \hat{j} ; \quad \bar{c}=\hat{i}-3 \hat{j}+5 \hat{k} ; \bar{d}=4 \hat{i}+\hat{j}-7 \hat{k} \)
\( \mathrm{P} \) \( \mathrm{d} \) is the shortest distance between the lines \( \mathrm{AB} \) and \( \mathrm{CD} \), then which of the following is True?
W
(A) \( \mathrm{d}=0 \), hence \( \mathrm{AB} \) and \( \mathrm{CD} \) intersect
(B) \( d=\frac{[\overrightarrow{\mathrm{AB}} \overrightarrow{\mathrm{CD}} \overrightarrow{\mathrm{BD}}]}{|\overrightarrow{\mathrm{AB}} \times \overrightarrow{\mathrm{CD}}|} \)
(C) \( \mathrm{AB} \) and \( \mathrm{CD} \) are skew lines and \( \mathrm{d}=\frac{23}{13} \)
(D) \( d=\frac{[\overrightarrow{\mathrm{AB}} \overrightarrow{\mathrm{CD}} \overrightarrow{\mathrm{AC}}]}{|\overrightarrow{\mathrm{AB}} \times \overrightarrow{\mathrm{CD}}|} \)
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