Find `vecB` at the origin due to the long wire carrying a current I

Explanation

We divide the wire into four parts 1, 2, 3 and 4. The induction at O is
`vecB=vecB_(1)+vecB_(2)+vecB_(3)+vecB_(4)` ….(i)
`vecB_(1)=0`
`vecB_(2)=vecB_(3)=(mu_(0)i)/(4pia)(costheta_(1)+costheta_(2))hatj`
`=(mu_(0)i)/(4pia)(cos90^(@)+1/(sqrt2))hatj=(mu_(0)i)/(4sqrt2pia)hatj` .......(ii)
`vecB_(4)=(mu_(0)i)/(4pia)(cos90^(@)+cos0^(@))hatk=(mu_(0)i)/(4pia)hatk` ....(iii)
using the above equations, `vecB=(mu_(0)i)/(4pia)(sqrt2hatjihatk)`