The vectors `veca and vecb ` are non collinear. Find for what value of x the vectors `vecc=(x-2)veca+vecb and vecd=(2x+1) veca-vecb` are collinear.?

Explanation

Both the vectors `vecc and vecd` are non-zero as the cofficients of `vecb` in the both are non-zero.
Two vectors `vecc and vecd` are collinear if one of them is a linear multiple of the other. Therefore,
`" "vecd=lamdavecc`
or `" "(2x+1)veca-vecb=lamda{(x-2)veca+vecb}" "`(i)
or `" "{(2x+1)-lamda(x-2)}veca-(1+lamda)vecb=0`
The above expression is of the form `pveca+qvecb=0,` where `veca and vecb` are non-collinear, and hence we have `p=0 and q=0`. Therefore,
`" "2x+1-lamda(x-2)=0" "`(ii)
and `" "1+lamda=0" "`(iii)
From (iii), `lamda=-1`, and putting this value in (i), we get `x=(1)/(3)`
Alternate method :
`" "vecc=(x-2)veca+vecb and vecd = (2x+1)veca-vecb` are collinear.
If `(x-2)/(2x+1)=(1)/(-1)`, then `x=(1)/(3)`