JEE Mathematics/ Lakshmikanta Satapathy/ Three dimensional Geometry QA Part 1/ Previous CBSE board question on distance between parallel lines solved with the related concepts

1. Physics Helpline
L K Satapathy
3D Geometry QA 1

2. Physics Helpline
L K Satapathy
2 1( )b a a
d
b
 

Question : Find the distance between the lines L1 and L2 , whose equations are
ˆ ˆˆ ˆ ˆ ˆ2 4 (2 3 6 )r i j k i j k      ˆ ˆˆ ˆ ˆ ˆ3 3 5 (2 3 6 )r i j k i j k     and
Answer : The given equations are
ˆ ˆˆ ˆ ˆ ˆ2 4 (2 3 6 ) . . . (1)r i j k i j k     
ˆ ˆˆ ˆ ˆ ˆ3 3 5 (2 3 6 ) . . . (2)r i j k i j k     
Comparing with their vector equations , we get1r a b and  2r a b 
1
ˆˆ ˆ2 4 ,a i j k   2
ˆˆ ˆ3 3 5a i j k   ˆˆ ˆ2 3 6and b i j k  
The lines L1 and L2 are parallel .  The distance between them is given by
3 D Geometry QA 1

3. Physics Helpline
L K Satapathy
3 D Geometry QA 1
2 1( )b a a
d
b
 
 
Now 2 1
ˆ ˆˆ ˆ ˆ ˆ(3 1) (3 2) (5 4) 2a a i j k i j k         
2 1
ˆˆ ˆ
( ) 2 3 6
2 1 1
i j k
b a a   

ˆ ˆˆ ˆ ˆ ˆ( 3 6) (12 2) (2 6) 9 14 4i j k i j k          
2 1( )b a a    81 196 16 293  
293
7

ˆˆ ˆ2 3 6 4 9 36 49 7b i j k        ˆˆ ˆ2 3 6and b i j k  
[ Ans ]

4. Physics Helpline
L K Satapathy
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