Find the shortest distance between the given lines Find the shortest distance and the vector equation of the line - Maths - Three Dimensional Geometry

Question

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 ^ )

Shruti Tyagi · Accepted Answer

Dear student,

We know that,Shortest distance between the line is d=a2→-a1→
b1→×b2→b1→×b2→Step 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→×b2→b1→×b2→Here, a1=8i^-9j^+10k^a2=15i^+29j^+5k^b1=3i^-16j^+7k^b2=3i^+8j^-5k^Let us obtain a2→-a1→a2→-a1→=15i^+29j^+5k^-8i^-9j^+10k^=15i^+29j^+5k^-8i^+9j^-10k^=7i^+38j^-5k^Step 2:Next let us obtain b1→×b2→b1→×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

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 ^ )