The shortest distance between the lines \(\frac{x-1}{2}=\frac{y-2}{3}=\frac{z-1}{4}\) and \(\frac{x+2}{7}=\frac{y-2}{8}=…

\(\begin{aligned}
& A \equiv(1,2,1), C(-2,2,-1) \\
& \text { where } \hat{\mathrm{n}}=\left|\begin{array}{lll}
\hat{\mathrm{i}} & \hat{j} & \hat{k} \\
2 & 3 & 4 \\
7 & 8 & 2
\end{array}\right|=-26 \hat{\mathrm{i}}+24 \hat{\mathrm{j}}-5 \hat{\mathrm{k}} \\
& \\
& =\left|\frac{(3 \hat{\mathrm{i}}+2 \hat{k}) \cdot(26 \hat{\mathrm{i}}-24 \hat{\mathrm{j}}+5 \hat{k})}{\sqrt{1277}}\right| \\
& =\frac{78+10}{\sqrt{1277}}=\frac{88}{\sqrt{1277}}
\end{aligned}\)