1.

If veca, vecb, vecc non-zero vectors such that veca is perpendicular to vecb and vecc and |veca|=1, |vecb|=2, |vecc|=1, vecb.vecc=1. There is a non-zero vector coplanar with veca+vecb and 2vecb-vecc and vecd.veca=1, then the minimum value of |vecd| is

Answer»

`2/SQRT(13)`
`3/sqrt(3)`
`4/sqrt(5)`
`4/sqrt(13)`

Solution :`veca.vecb=veca.vecc=0, |veca|=|vecc|=1,|vecb|=2` and `vecb.vecc=1`
but `vecd.veca=1 rArr 1=x(1+0)+0 rArr x=1`
`rArr vecd=veca +vecb+y(2vecb-vecc)`
`rArr |vecd|^(2)=|veca|^(2)+|b|^(2)=2veca.vecb+y^(2)(2vecb-vecc)^(2)+2y(veca+vecb).(2vecb-vecc)`
`rArr =1+4+y^(2)(16+1-4)+2y(8-1)`
`=13y^(2)+14y+5`
`therefore |vecd|_("MIN" ) = sqrt((4.13.5-14.14)/(4.13))=4/sqrt(13)`


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