Find the shortest distance between the lines whose
vector equations are
= ( 
+ 2 
+ 3 
) + λ (2 
+ 3 
+ 4 
) and
= (2 
+ 4 
+ 5 
) + μ (4 
+ 6 
+ 8 
)
Text Solution
Verified by ExpertsCHECK THE SOLUTION.
Sol. The given lines are
= ( 
+ 2 
+ 3 
) + λ (2 
+ 3 
+ 4 
) …(i)
and 
= (2 
+ 4 
+ 5 
) + 2 μ (2 
+ 3 
+ 4 
) …(ii)
Equation (ii) can rewritten as
= (2 
+ 4 
+ 5 
) + μ′ (2 
+ 3 
+ 4 
) …(iii)
Where μ′ = 2 μ
These two lines passes through the points having
position vectors
= 
+ 2 
+ 3 
and 
= 2 
+ 4 
+ 5 
respectively and both are parallel to the vector 
= 2 
+ 3 
+ 4 
∴ Shortest distance = 
…(iv)
We have ( 
– 
) × 
= ( 
+ 2 
+ 2 
) × (2 
+ 3 
+ 4 
) = 
= (8 – 6) 
– (4 – 4) 
+ (3 – 4) 
= 2 
– 0 
– 
∴ | ( 
) × 
| = 
= 
and | 
| = 
= 
Substituting the values of | ( 
) × 
| and | 
| in (iv), we get
∴ shortest distance = 
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