The shortest distance between the skew linesl1:r⃗ =a⃗ 1+λb⃗ 1l2:r⃗ =a⃗ 2+μb⃗ 2 is

[a] Let PQ be the shortest distance vector between l1 and l2. Now, l1 passes through A1(a⃗ 1) and is parallel to b⃗ 1 and l2 passes through A2(a⃗ 2) and is parallel to b⃗ 2. Since, PQ is perpendicular to both l1 and l2 is is parallel to b⃗ 1×b⃗ 2. Let n^ be the unit vector along PQ. Then, n^=b⃗ 1×b⃗ 2∣∣b⃗ 1×b⃗ 2∣∣ Let d be the shortest distance between the given lines l1 andl2. ∣∣PQ−→−−∣∣=d and PQ−→−−=dn^. Next PQ being the line of shortest distance between l1 and l2 is the projection of the line joining the points A1(a⃗ 1) and A2(a⃗ 2)onn^. ∣∣PQ−→−−∣∣=∣∣A⃗ 1A⃗ 2.n^∣∣⇒d=∣∣∣∣(a⃗ 2−a⃗ 1).b⃗ 1×b⃗ 2∣∣b⃗ 1×b⃗ 2∣∣∣∣∣∣