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Line and Plane — Maths STD 12 Science — Question

Maharashtra BoardEnglish MediumSTD 12 ScienceMathsLine and Plane4 Marks
Question
Prove the Theorem : The distance between lines $\bar{r}=\bar{a}_1+\lambda_1 \bar{b}_1$ and $\bar{r}=\bar{a}_2+\lambda_2 \bar{b}_2$ is $\left|\frac{\left(\bar{a}_2-\bar{a}_1\right) \cdot \bar{b}_1 \times \bar{b}_2}{\left|\bar{b}_1 \times \bar{b}_2\right|}\right|$
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Answer

Proof: Let $\mathrm{L}_1$ and $\mathrm{L}_2$ be the lines whose equations are $\bar{r}=\bar{a}_1+\lambda_1 \bar{b}_1, \bar{r}=\bar{a}_2+\lambda_2 \bar{b}_2$ respectively. Let $P Q$ be the segment which is perpendicular to both $L_1$ and $L_2$.
To find the length of segment PQ.
Lines $L_1$ and $L_2$ pass through pointsA $\left(\bar{a}_1\right)$ and $\mathrm{B}\left(\bar{a}_2\right)$ respectively. Lines $\mathrm{L}_1$ and $\mathrm{L}_2$ are parallel to $\bar{b}_1$ and $\bar{b}_2$ respectively.
As PQ is perpendicular to both $\mathrm{L}_1$ and $\mathrm{L}_2$, it is parallel to $\bar{b}_1 \times \bar{b}_2$
The unit vector along $\overline{P Q}=$ unit vector along $\bar{b}_1 \times \bar{b}_2=\hat{n}$ (say) $\mathrm{PQ}=$ The projection of $\overline{A B}$ on $\overline{P Q}=\overline{A B} \cdot \hat{n}$
$
P Q=\left|\frac{\left(\bar{a}_2-\bar{a}_1\right) \cdot\left(\bar{b}_1 \times \bar{b}_2\right)}{\left|\bar{b}_1 \times \bar{b}_2\right|}\right|
$

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