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Question

Find the shortest distance between the lines r=^i+2^j+3^k+λ(2^i+3^j+4^k) and r=2^i+4^j+5^k+μ(4^i+6^j+8^k).

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Solution

Let l1 and l2 be the given lines whose equations are
r=a1+λb
and r=a2+2μb

Through the points a1=^i+2^j+3^k and a2=2^i+4^j+5^k are parallel to b=2^i+3^j+4^k.

Hence the distance between the lines using the formula :
|b×(a2a1)||b|=|(2^i+3^j+4^k)×(^i+2^j+2^k)||b|

=|(2^i+3^j+4^k)×(^i+2^j+2^k)||2^i+3^j+4^k|

=∣ ∣ ∣^i^j^k234122∣ ∣ ∣4+9+16=|2^i+^k|29

Ans. =529

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