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Question

Find the shortest distance between the following pair of lines.
¯r=(¯i+2¯j+¯k)+λ(2¯i¯j+3¯k) & ¯r=(¯i3¯j¯k)+μ(3¯i+2^j5¯k).

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Solution

Accordingtoquestion,wehavetwopairofline..........¯r1=(¯i+2¯j+¯k)+λ(2¯i¯j+3¯k)¯r2=(¯i3¯j¯k)+μ(3¯i+2^j5¯k)Now,
Comparingthegivenequnwiththeequations:r1=a1+λb1andr2=a2+λb2
a1=ˆi2ˆj+ˆka2=ˆi3ˆjˆkb1=2ˆiˆj+3ˆkb2=3ˆi+2ˆj5ˆka2a1=(ˆi3ˆjˆk)(ˆi2ˆj+ˆk)=ˆi3ˆjˆkˆi+2ˆjˆk=(ˆj2ˆk)(i)Nowb1×b2=∣ ∣ ∣iˆˆjˆk213325∣ ∣ ∣=ˆi(56)ˆj(109)+ˆk(4+3)=ˆi+19ˆj+7ˆk(ii)b1×b2=(1)2+(19)2+(7)2=1+361+49=411=21(a2a1)(b1×b2)=(ˆj2ˆk)(ˆi+19ˆj+7ˆk)=1+1914=6andshortestdistancebetweentheline:d=∣ ∣(a2a1)(b1×b2)b1×b2∣ ∣=621=27sothatthecorrectsolution27


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