The equations of the given lines are
→r=(^i+2^j+3^k)+λ(^i+^j+^k)
→r=(2^i−^j−^k)+μ(−^i+^j−^k)
It is known that the shortest distance between the lines →r=→a1+λ→b1 and →r=→a2+μ→b2 is given by
d=∣∣
∣
∣∣(→b1×→b2).(→a2−→a1)∣∣→b1×→b2∣∣∣∣
∣
∣∣ ......(1)
Comparing the equations,
→a1=^i+2^j+3^k
→a2=2^i−^j−^k
→b1=^i+^j+^k
→b2=−^i+^j−^k
→a2−→a1=2^i−^j−^k−^i−2^j−3^k=^i−3^j−4^k
→b1×→b2=∣∣
∣
∣∣^i^j^k111−11−1∣∣
∣
∣∣
=^i(−1−1)−^j(−1+1)+^k(1+1)
=−2^i+2^k
∣∣→b1×→b2∣∣=√(−2)2+22=2√2
∴d=∣∣
∣
∣∣(→b1×→b2).(→a2−→a1)∣∣→b1×→b2∣∣∣∣
∣
∣∣
=∣∣
∣
∣∣(−2^i+2^k).(^i−3^j−4^k)2√2∣∣
∣
∣∣
=∣∣∣−2−82√2∣∣∣
=5√2
∴ the shortest distance between the two lines is 5√2