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The shortest distance between the skew lines $\hat{\mathbf{r}}=(\hat{\mathbf{i}}+2 \hat{\mathbf{j}}+3…
The shortest distance between the skew lines $\hat{\mathbf{r}}=(\hat{\mathbf{i}}+2 \hat{\mathbf{j}}+3 \hat{\mathbf{k}})+t(\hat{\mathbf{i}}+3 \hat{\mathbf{j}}+2 \hat{\mathbf{k}})$ and $\hat{\mathbf{r}}=(4 \hat{\mathbf{i}}+5 \hat{\mathbf{j}}+6 \hat{\mathbf{k}})+t(2 \hat{\mathbf{i}}+3 \hat{\mathbf{j}}+\hat{\mathbf{k}})$ is
$sqrt{6}$ $3$ $2 \sqrt{3}$ $sqrt{3}$ Solution
Given skew lines are
$
\mathbf{r}=(\hat{\mathbf{i}}+2 \hat{\mathbf{j}}+3 \hat{\mathbf{k}})+t(\hat{\mathbf{i}}+3 \hat{\mathbf{j}}+2 \hat{\mathbf{k}})
$
and $\mathrm{r}=(4 \hat{\mathbf{i}}+5 \hat{\mathbf{j}}+6 \hat{\mathbf{k}})+t(2 \hat{\mathbf{i}}+3 \hat{\mathbf{j}}+\hat{\mathbf{k}})$
Here,
$
\begin{aligned}
& \mathbf{a}_1=\hat{\mathbf{i}}+2 \hat{\mathbf{j}}+3 \hat{\mathbf{j}} \\
& \mathrm{b}_1=\hat{\mathbf{i}}+3 \hat{\mathbf{j}}+2 \hat{\mathbf{k}} \\
& \mathrm{a}_2=4 \hat{\mathbf{i}}+5 \hat{\mathbf{j}}+6 \hat{\mathbf{k}} \\
& \mathrm{b}_2=2 \hat{\mathbf{i}}+3 \hat{\mathbf{j}}+\hat{\mathbf{k}}
\end{aligned}
$
and
$
\mathbf{b}_2=2 \hat{\mathbf{i}}+3 \hat{\mathbf{j}}+\hat{\mathbf{k}}
$
Now, 0
$
\begin{aligned}
\mathbf{a}_2-\mathbf{a}_1 & =4 \hat{\mathbf{i}}+5 \hat{\mathbf{j}}+6 \hat{\mathbf{k}}-(\hat{\mathbf{i}}+2 \hat{\mathbf{j}}+3 \hat{\mathbf{k}}) \\
& =3 \hat{\mathbf{i}}+3 \hat{\mathbf{j}}+3 \hat{\mathbf{k}}
\end{aligned}
$
and
$
\begin{aligned}
\mathrm{b}_1 \times \mathrm{b}_2 & =\left|\begin{array}{ccc}
\hat{\mathbf{i}} & \hat{\mathbf{j}} & \hat{\mathbf{k}} \\
1 & 3 & 2 \\
2 & 3 & 1
\end{array}\right| \\
& =\hat{\mathbf{i}}(3-6)-\hat{\mathbf{j}}(1-4)+\hat{\mathbf{k}}(3-6) \\
& =-3 \hat{\mathbf{i}}+3 \hat{\mathbf{j}}-3 \hat{\mathbf{k}}
\end{aligned}
$
$\therefore$ The shortest distance between skew lines
$
\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{|(3 \hat{\mathbf{i}}+3 \hat{\mathbf{j}}+3 \hat{\mathbf{k}}) \cdot(-3 \hat{\mathbf{i}}+3 \hat{\mathbf{j}}-3 \hat{\mathbf{k}})|}{\sqrt{(-3)^2+(3)^2+(-3)^2}} \\
& =\frac{|-9+9-9|}{\sqrt{9+9+9}}=\frac{9}{3 \sqrt{3}}=\sqrt{3}
\end{aligned}
$
Asked in: AP EAMCET 2014
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