Given : point A ≡ (2, 3, 2)
Equation of line :
![](https://www.sarthaks.com/?qa=blob&qa_blobid=10357086214276456888)
To Find : i) equation of line
ii) distance d
Formulae :
1. Equation of line :
Equation of line passing through point A (a1, a2, a3) and parallel to vector
![](https://www.sarthaks.com/?qa=blob&qa_blobid=12389860323071638650)
is given by
\(\vec r=\bar a_1+\lambda \bar{b}\)
Where,
![](https://www.sarthaks.com/?qa=blob&qa_blobid=7774893256733554014)
![](https://www.sarthaks.com/?qa=blob&qa_blobid=18033482932973399315)
4. Shortest distance between two parallel lines :
The shortest distance between the parallel lines \(\vec r=\bar a_1+\lambda \bar{b}\) and
\(\vec r=\bar a_2+\lambda \bar {b}\) is given by,
![](https://www.sarthaks.com/?qa=blob&qa_blobid=1326548759434620150)
As the required line is parallel to the line
![](https://www.sarthaks.com/?qa=blob&qa_blobid=10357086214276456888)
Therefore, the vector parallel to the required line is
![](https://www.sarthaks.com/?qa=blob&qa_blobid=14704748535294999691)
Therefore, equation of line passing through A and parallel to \(\vec b\) is
![](https://www.sarthaks.com/?qa=blob&qa_blobid=4862754303813317947)
Now, to calculate distance between above line and given line,
![](https://www.sarthaks.com/?qa=blob&qa_blobid=6075757118034096527)
Therefore, the shortest distance between the given lines is
![](https://www.sarthaks.com/?qa=blob&qa_blobid=12406740347275379306)