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vec a , vec b , vec c are three coplanar...

` vec a , vec b , vec c` are three coplanar unit vectors such that ` vec a+ vec b+ vec c=0.` If three vectors ` vec p , vec q ,` and `vec r` are parallel to ` vec a , vec b ,` and `vec c ,` respectively, and have integral but different magnitudes, then among the following options, `| vec p+ vec q+ vec r|` can take a value equal to

A

(a) 1

B

(b) 0

C

`(c) sqrt(3)`

D

(d) 2

Text Solution

Verified by Experts

The correct Answer is:
C, D
Let a,b and c lie in the XY-plane
Let `a=hati,b=-(1)/(2)hati+(sqrt(3))/(2)hatj and c=-(1)/(2)hati-(sqrt(3))/(2)hatj`
Therefore, `|p+q+r|=|lamda a+mub+vc|`
`=|lamdahati+mu(-(1)/(2)hati+(sqrt(3))/(2)hatj)+v(-(1)/(2)hati-(sqrt(3))/(2)hatj)|`
`|(lamda+(mu)/(2)-(v)/(2))hati+(sqrt(3))/(2)(mu-v)hatj|`
`=sqrt((lamda-(mu)/(2)-(v)/(2))^(2)+(3)/(4)(mu-v)^(2))`
`=sqrt(lamda^(2)+mu^(2)+v^(2)-lamdamu-lamdav-muv)`
`=(1)/(sqrt(2))sqrt((lamda-mu)^(2)+(mu-v)^(2)+(v-lamda)^(2))`
`=(1)/(sqrt(2))sqrt(1+1+4)=sqrt(3)`.
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