Question 35 (Choice 2)  CBSE Class 12 Sample Paper for 2024 Boards  Solutions of Sample Papers and Past Year Papers  for Class 12 Boards
Last updated at April 16, 2024 by Teachoo
An aeroplane is flying along the line
r
=λ(
i
ˆ

j
ˆ
+
k
ˆ
); where ' λ ' is a scalar and another aeroplane is flying along the line r =i ˆj ˆ+μ(2j ˆ+
k
ˆ
); where '
μ
' is a scalar. At what points on the lines should they reach, so that the distance between them is the shortest? Find the shortest possible distance between them.
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ο»ΏThe given lines are nonparallel lines.
Let Shortest distance = (ππ) β 
Since (ππ) β is shortest distance,
(ππ) β β₯ Line 1
(ππ) β β₯ Line 2
Point P
Since point P lies on Line 1
Position vector of P
= π(π Λβπ Λ+π Λ)
= (π) π Μ+(βπ)π Μ+(π)π Μ
Now,
(π·πΈ) β = Position vector of Q β Position vector of P
= [π Μ+(β1β2π)π Λ+(π)π Μ ]β[(π) π Μ+(βπ)π Μ+(π)π Μ]
= ο»Ώ(1 β π)π€Μ + (β 1 β 2π + π)π₯Μ + (π β π)π Μ
Now,
(π·πΈ) β β₯ Line 1 (π β = π(π Λβπ Λ+π Λ) )
Thus,
(ππ) β β₯ (π Μβπ Μ+π Μ )
And
(π·πΈ) β . (π Μβπ Μ+π Μ )=π
ο»Ώ(1 β π)π€Μ + (β 1 β 2π + π)π₯Μ + (π β π)π Μ. (π Μβπ Μ+π Μ )=π
(1β π)1 + (β1 β2π + π)(β1) + (π β π)1 = 0
(1β π) + (1 + 2π  π) + (π β π) = 0
(1 + 1) + (β π β π β π) + (2π + π) = 0
2 β 3π + 3π = 0
3π β 3π = β 2
Similarly
(π·πΈ) β β₯ Line 2 (π β = π Λβπ Λ+π(β2π Λ+π Λ))
Thus,
(ππ) β β₯(β2π Λ+π Λ)
And
(π·πΈ) β .(β2π Λ+π Λ)=π
(1 β π)π€Μ + (β 1 β 2π + π)π₯Μ + (π β π)π Μ.(β2π Λ+π Λ )=0
(1 β π)(0) + (β 1 β 2π + π)(β2) + (π β π)(1) = 0
0 + (2 +4π β 2π) + (π β π) = 0
(2) + (β2π β π) + (4π + π) = 0
2 β 3π + 5π = 0
5π β 3π = β2
Thus, our equations are
3π β 3π = β2 β¦(1)
5π β 3π = β2 β¦(2)
Solving (1) and (2)
We get
π = 0, π = π/π
Point P
Position vector of P
= (π) π Μ+(βπ)π Μ+(π)π Μ
Putting π = π/π
= π/π π Μβπ/π π Μ+ π/π π Μ
Point Q
Position vector of Q
= π Μ+(β1β2π)π Λ+(π)π Μ
Putting π = 0
=( π) Μ+(β1 β0) π Μ+(0)π Μ
= π Μβπ Μ
Now,
(π·πΈ) β = Position vector of Q β Position vector of P
=[ π Μβ(π]) Μβ[ π/π π Μβπ/π π Μ+ π/π π Μ]
= π/π π Μβ(π )/π π Μβπ/π π Μ
And,
Shortest distance = (ππ) β 
= β((1/3)^2+(β1/3)^2+(2/3)^2 )
=β( 1/9+1/9 +4/9)
= β(6/9)
= β(π/π) units
Made by
Davneet Singh
Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 14 years. He provides courses for Maths, Science and Computer Science at Teachoo
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