The given lines are $\mathbf{r}=\mathbf{a}_1+\lambda \mathbf{b}_1, \mathbf{r}=\mathbf{a}_2+\mu \mathbf{b}_2$ where,
$$
\begin{aligned}
& \mathbf{a}_1=3 \mathbf{i}+5 \mathbf{j}+7 \mathbf{k}, \mathbf{b}_1=\mathbf{i}+2 \mathbf{j}+\mathbf{k} \\
& \mathbf{a}_2=-\mathbf{i}-\mathbf{j}-\mathbf{k}, \mathbf{b}_2=7 \mathbf{i}-6 \mathbf{j}+\mathbf{k} \\
& \quad\left|\mathbf{b}_1 \times \mathbf{b}_2\right|=\left|\begin{array}{ccc}
\mathbf{i} & \mathbf{j} & \mathbf{k} \\
1 & 2 & 1 \\
7 & -6 & 1
\end{array}\right| \\
\Rightarrow \quad & |\mathbf{i}(2+6)-\mathbf{j}(1-7)+\mathbf{k}(-6-14)| \\
\Rightarrow & |8 \mathbf{i}+6 \mathbf{j}-20 \mathbf{k}| \\
\Rightarrow & \sqrt{64+36+400}=\sqrt{500}=10 \sqrt{5}
\end{aligned}
$$
Now,
$$
\begin{aligned}
& {\left[\left(\mathbf{a}_2-\mathbf{a}_1\right) \mathbf{b}_1 \mathbf{b}_2\right]=\left(\mathbf{a}_2-\mathbf{a}_1\right) \cdot\left(\mathbf{b}_1 \times \mathbf{b}_2\right)} \\
& =(-4 \mathbf{i}-6 \mathbf{j}-8 \mathbf{k}) \cdot(8 \mathbf{i}+6 \mathbf{j}-20 \mathbf{k}) \\
& =-32-36+160 \\
& =160-68=92
\end{aligned}
$$
Shortest distance
$$
\begin{aligned}
& =\frac{\left[\left(\mathbf{a}_2-\mathbf{a}_1\right) \cdot\left(\mathbf{b}_1 \times \mathbf{b}_2\right)\right]}{\left|\mathbf{b}_1 \times \mathbf{b}_2\right|} \\
& =\frac{92}{10 \sqrt{5}}=\frac{46}{5 \sqrt{5}}
\end{aligned}
$$