Given skew line

$$ \mathbf{r}=(2 \hat{\mathbf{i}}-\hat{\mathbf{j}})+t(\hat{\mathbf{i}}+2 \hat{\mathbf{k}}) \Rightarrow \mathbf{r}=(-2 \hat{\mathbf{i}}+\hat{\mathbf{k}})+s(\hat{\mathbf{i}}-\hat{\mathbf{j}}-\hat{\mathbf{k}}) $$

We know that shortest distance between lines vector equations. $\mathbf{r}=\mathbf{a}_1+t \mathbf{b}_1 \ldots$ and $\mathbf{r}=\mathbf{a}_2+s \mathbf{b}_2 \ldots$ is

$$ \left|\frac{\left(\mathbf{b}_1 \times \mathbf{b}_2\right) \cdot\left(\mathbf{a}_2-\mathbf{a}_1\right)}{\left(\mathbf{b}_1 \times \mathbf{b}_2\right)}\right| $$

Now, $\mathbf{r}=(2 \hat{\mathbf{i}}-\hat{\mathbf{j}})+t(\hat{\mathbf{i}}+2 K)$ comparing with

$$ \mathbf{r}=\mathbf{a}_1+t \mathbf{b}_1 \Rightarrow \mathbf{a}_1=2 \hat{\mathbf{i}}-\hat{\mathbf{j}}, \mathbf{b}_1=\hat{\mathbf{i}}+2 \hat{\mathbf{k}} $$

and $\mathbf{r}=(-2 \hat{\mathbf{i}}+\hat{\mathbf{k}})+s(\hat{\mathbf{i}}-\hat{\mathbf{j}}-\hat{\mathbf{k}})$ comparing with

$$ \begin{aligned} \mathbf{r} & =\mathbf{a}_2+t \mathbf{b}_2 \\ \therefore \quad \mathbf{a}_2 & =-2 \hat{\mathbf{i}}+\hat{\mathbf{k}}, \mathbf{b}_2=\hat{\mathbf{i}}-\hat{\mathbf{j}}-\hat{\mathbf{k}} \end{aligned} $$

Now, $\left(\mathbf{a}_2-\mathbf{a}_1\right)=(-2 \hat{\mathbf{i}}+\hat{\mathbf{k}}-2 \hat{\mathbf{i}}+\hat{\mathbf{j}})=-4 \hat{\mathbf{i}}+\hat{\mathbf{j}}+\hat{\mathbf{k}}$

$$ \mathbf{b}_1 \times \mathbf{b}_2=\left|\begin{array}{ccc} \hat{\mathbf{i}} & \hat{\mathbf{j}} & \hat{\mathbf{k}} \\ 1 & 0 & 2 \\ 1 & -1 & -1 \end{array}\right|=\hat{\mathbf{i}}(2)-\hat{\mathbf{j}}(-1-2)+\hat{\mathbf{k}}(-1)=2 \hat{\mathbf{i}}+3 \mathbf{j} \cdot \hat{\mathbf{i}} $$

$$ \begin{aligned} &\left|\mathbf{b}_1 \times \mathbf{b}_2\right|=\sqrt{2^2+3^2+1^2}=\sqrt{4+9+1}=\sqrt{14}\\ &\begin{aligned} \text { So, shortest distance } & =\left|\frac{(2 \hat{\mathbf{i}}+3 \hat{\mathbf{j}}-\hat{\mathbf{k}})(-4 \hat{\mathbf{i}}+\hat{\mathbf{j}}+\hat{\mathbf{k}})}{\sqrt{14}}\right| \\ & =\left|\frac{-8+3-1}{\sqrt{14}}\right|=\frac{6}{\sqrt{14}}=\frac{3 \sqrt{2}}{\sqrt{7}} \end{aligned}\\ &\text { Hence, the shortest distance between skew lines is } \frac{3 \sqrt{2}}{\sqrt{7}} \end{aligned} $$

Three Dimensional Geometry

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