An arc `AC` of a circle subtends a right angle at then the center `O`. the point B divides the are in the ratio `1:2`, If `vecOA = a & vecOB = b`. then the vector `vecOC` in terms of `veca &vec b`, is

Explanation

1. vector `vecc` is coplanar with vector `veca and vecb`. Therefore,
`vecc=xveca+yvecb`
Point B didvides arc AC in the ratio 1:2 so that `angleAOB=30^(@) and angleBOC = 60^(@)`
we have to find the values of x and y when we arew given
`|veca|=|vecb|=|vecc|=r (say).`
`veca.vecb=r^(2)cos30^(@)=r^(2)sqrt3/2and veca.vecb=0`
`vecb.vecc=r^(2)cos 60^(@)=r^(2)/2`
Multiplying both sides of (i) scalarly by `vecc and veca,vecc.vecc=xveca.vecc+yvecb.vecc`
and `vecc.veca=xveca.veca+yvecb.veca`
`r^(2)=0+r^(2)/2y,y=2`
`and 0=xr^(2)+yr^(2)sqrt3/2`
putting `y=2,x=-sqrt3`
`vecc=-sqrt3veca+2vecb`
`vecc=-sqrt3veca+2vecb`