Find the shortest distance between the following lines : \[\vec{r}=3 \hat{i}+5 \hat{j}+7 \hat{k}+\lambda(\hat{i}-2 \hat{j}+\hat{k}) \text { and } \vec{r}=(-\hat{i}-\hat{j}-\hat{k})+\mu(7 \hat{i}-6 \hat{j}+\hat{k}) . \]

$\displaystyle \vec{a}_1 = 3\hat{i} + 5\hat{j} + 7\hat{k} \qquad \vec{b}_1 = (\hat{i} - 2\hat{j} + \hat{k})$$\displaystyle \vec{a}_2 = -\hat{i} - \hat{j} - \hat{k} \qquad \vec{b}_2 = 7\hat{i} - 6\hat{j} + \hat{k}$$\displaystyle (\vec{a}_2 - \vec{a}_1) = -4\hat{i} - 6\hat{j} - 8\hat{k}, \qquad \vec{b}_1 \times \vec{b}_2 = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ 1 & -2 & 1 \\ 7 & -6 & 1 \end{vmatrix} = 4\hat{i} + 6\hat{j} + 8\hat{k}$$\displaystyle (\vec{a}_2 - \vec{a}_1) \cdot (\vec{b}_1 \times \vec{b}_2) = -16 - 36 - 64 = -116$$\displaystyle |\vec{b}_1 \times \vec{b}_2| = \sqrt{16 + 36 + 64} = \sqrt{116}$S.D. $\displaystyle = \dfrac{|-116|}{\sqrt{116}} = \sqrt{116}$