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 The resultant of two vectors u and v ...
 Entrance Exam
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Thu November 01, 2012 By: Pramod R
The resultant of two vectors u and v is perpendicular to u and is half of the magnitude of v. Then what is the angle between u and v ? Please explain.
Expert Reply
Resultant of 2 vectors 'u' and 'v' is given as
=?{uÂ²+vÂ²+2uvcos?}
Hence
v/2 =?{uÂ²+vÂ²+2uvcos?}
?vÂ² =4{uÂ²+vÂ²+2uvcos?}.....(i)
Angle '?' of resultant with 'u' is given by
tan? =(vsin?)/(u+vcos?)
Since ?=90Â°
u+vcos?=0 i.e u=?vcos?....(ii)
from (i) and (ii), we get
vÂ² =4{(?vcos?)Â²+vÂ²+2(?vcos?)vcos?}
? vÂ² =4{vÂ²cosÂ²?)+vÂ²?2vÂ²cosÂ²?
?vÂ² =4vÂ²{1?cosÂ²?}
i.e 1=4?4cosÂ²?}
cos cosÂ²?=3/4 or
cos?=Â±?3/2 or
Since the resultant is less than 'u' hence angle ?> 90Â°.
Hence cos?=??3/2 and ?=150Â°
=?{uÂ²+vÂ²+2uvcos?}
Hence
v/2 =?{uÂ²+vÂ²+2uvcos?}
?vÂ² =4{uÂ²+vÂ²+2uvcos?}.....(i)
Angle '?' of resultant with 'u' is given by
tan? =(vsin?)/(u+vcos?)
Since ?=90Â°
u+vcos?=0 i.e u=?vcos?....(ii)
from (i) and (ii), we get
vÂ² =4{(?vcos?)Â²+vÂ²+2(?vcos?)vcos?}
? vÂ² =4{vÂ²cosÂ²?)+vÂ²?2vÂ²cosÂ²?
?vÂ² =4vÂ²{1?cosÂ²?}
i.e 1=4?4cosÂ²?}
cos cosÂ²?=3/4 or
cos?=Â±?3/2 or
Since the resultant is less than 'u' hence angle ?> 90Â°.
Hence cos?=??3/2 and ?=150Â°
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