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Question
Let (^i+^j+^k),(2^i+5^j),(3^i+2^j−3^k) and (^i−6^j−^k) be the position vectors of points of points A,B,C,D respectively. Then show that AB∥CD
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Solution
¯OA=^i+^j+^k;→OB=2^i+5^j;→OC=3^i+2^j−3^k;→OD=^i−6^j−^k→AB=→OB−→OA=(2^i+5^j)−(^i+^j+^k)=^i+4^j−^k→CD=→OD−→OC=(^i−6^i−^k)−(3^i+2^j−3^k)=−2^i−8^j+2^k=−2[^i+4^j−^k]=−2(¯AB)⇒→CD=−2(→AB)⇒ AB and CD are parallel
¯OA=^i+^j+^k;→OB=2^i+5^j;→OC=3^i+2^j−3^k;→OD=^i−6^j−^k
→AB=→OB−→OA=(2^i+5^j)−(^i+^j+^k)=^i+4^j−^k
→CD=→OD−→OC=(^i−6^i−^k)−(3^i+2^j−3^k)=−2^i−8^j+2^k
=−2[^i+4^j−^k]
=−2(¯AB)
⇒→CD=−2(→AB)
⇒ AB and CD are parallel
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