Find the shortest distance between the given lines.
Find the shortest distance and the vector equation of the line of shortest distance between the lines given by:
  r = ( 8 + 3 λ ) i ^   - ( 9 + 16 λ ) j ^ + ( 10 + 7 λ ) k ^   a n d   r = 15 i ^ + 29 j ^ + 5 k ^   + μ ( 3 i ^ + 8 j ^ - 5 k ^ )

Dear student,

We know that,Shortest distance between the line is d=a2-a1.b1×b2b1×b2Step 1:-The given lines areL1: r=8+3λi^-9+16λj^+10+7λk^L2: r=15i^+29j^+5k^+μ3i^+8j^-5k^Let L1 can be written asr=8i^-9j^+10k^+λ3i^-16j^+7k^The shortest distance between the line isd=a2-a1.b1×b2b1×b2Here, a1=8i^-9j^+10k^a2=15i^+29j^+5k^b1=3i^-16j^+7k^b2=3i^+8j^-5k^Let us obtain a2-a1a2-a1=15i^+29j^+5k^-8i^-9j^+10k^=15i^+29j^+5k^-8i^+9j^-10k^=7i^+38j^-5k^Step 2:Next let us obtain b1×b2b1×b2=i^j^k^3-16738-5=i^80-56-j^-15-21+k^24+48=24i^+36j^+72k^=122i^+3j^+6k^b1×b2=22+32+62=4+9+36=7Step 3:Now substituting the respective values we get,d=7i^+38j^-5k^.24i^+36j^+72k^84=168+1368-36084=117684=14Hence the shortest distance between the line is 14

Regards
 

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