HSC Science (Computer Science) 12th Board ExamMaharashtra State Board
Account
It's free!

User


Login
Create free account


      Forgot password?
Share
Notifications

View all notifications
Books Shortlist
Your shortlist is empty

Solution - Find the Shortest Distance Between the Lines r=(4i-j)+λ(i+2j-3k) and r=(i-j+2k)+μ(i+4j-5k) - HSC Science (Computer Science) 12th Board Exam - Mathematics and Statistics

Question

Find the shortest distance between the lines

`bar r = (4 hat i - hat j) + lambda(hat i + 2 hat j - 3 hat k)`

and

`bar r = (hat i - hat j + 2 hat k) + mu(hat i + 4 hat j -5 hat k)`

where λ and μ are parameters

 

Solution

We know that the shortest distance between the lines 

`bar r=bara_1+lambdabarb_1 and barr=bara_2+mubarb_2 `

Given equation of lines are

`bar r=(4hati-hatj)+lambda(hati+2hatj-3hatk) and barr=(hati-hatj+2hatk)+mu(hati+4hatj-5hatk)`

`therefore bara_2-bara_1=(hati-hatj+2hatk)-(4hati-hatj)=-3hati+2hatk`

`barb_1xxbarb_2=|[hati,hatj,hatk],[1,2,-3],[1,4,-5]|`

`=hati(-10+12)-hatj(-5+3)+hatk(4-2)`

`=2hati+2hatj+2hatk`

`(bara_2-bara_1).(barb_1xxbarb_2)=(-3hati+2hatk).(2hati+2hatj+2hatk)=-3xx2+0xx2+2xx2=-2`

`|barb_1xxbarb_2|=sqrt(4+4+4)=2sqrt3`

Shortest distance =

`d=|-2/(2sqrt3)|=|-1/sqrt3|=1/sqrt3 units.`

 

 

 

Is there an error in this question or solution?

APPEARS IN

 2015-2016 (March) (with solutions)
Question 2.1.3 | 3 marks

Reference Material

Solution for question: Find the Shortest Distance Between the Lines r=(4i-j)+λ(i+2j-3k) and r=(i-j+2k)+μ(i+4j-5k) concept: null - Shortest Distance Between Two Lines. For the courses HSC Science (Computer Science), HSC Science (General) , HSC Arts, HSC Science (Electronics)
S