Let A(→a),B(→b),C(→c) and D(→d) be four points such that →a=−2^i+4^j+3^k,→b=2^i−8^j,→c=^i−3^j+5^k,→d=4^i+^j−7^k. If m is the shortest distance between the lines AB and CD, then which of the following is (are) CORRECT?
A
m=0, hence AB and CD intersect
No worries! We‘ve got your back. Try BYJU‘S free classes today!
B
m=[−−→AB−−→CD−−→BD]∣∣∣−−→AB×−−→CD∣∣∣
Right on! Give the BNAT exam to get a 100% scholarship for BYJUS courses
C
AB and CD are skew lines and m=2313
Right on! Give the BNAT exam to get a 100% scholarship for BYJUS courses
D
m=[−−→AB−−→CD−−→AC]∣∣∣−−→AB×−−→CD∣∣∣
Right on! Give the BNAT exam to get a 100% scholarship for BYJUS courses
Open in App
Solution
The correct options are Bm=[−−→AB−−→CD−−→BD]∣∣∣−−→AB×−−→CD∣∣∣ CAB and CD are skew lines and m=2313 Dm=[−−→AB−−→CD−−→AC]∣∣∣−−→AB×−−→CD∣∣∣ Given, →a=−2^i+4^j+3^k;→b=2^i−8^j→c=^i−3^j+5^k;→d=4^i+^j−7^k−−→AB=→b−→a=4^i−12^j−3^k−−→CD=→d−→c=3^i+4^j−12^k−−→AC=→c−→a=3^i−7^j+2^k−−→BD=→d−→b=2^i+9^j−7^k
By definition, m=(−−→AB×−−→CD)⋅−−→AC|−−→AB×−−→CD|⋯(1) =(−−→AB×−−→CD)⋅−−→BD|−−→AB×−−→CD|⋯(2)
−−→AB×−−→CD=13(12^i+3^j+4^k)∴|−−→AB×−−→CD|=13√144+9+16=169 ∴m=13(12^i+3^j+4^k)169⋅(3^i−7^j+2^k) [ Using (1)] =13169[36−21+8] =2313
Also, m=13(12^i+3^j+4^k)⋅(2^i+9^j−7^k)169 [ Using (2) ] =13169[24+27−28] =2313