Let ` vec a , vec b, vec c` be three non-zero vectors such that any two of them are non-collinear. If ` vec a+2 vec b` is collinear with ` vec ca n d vec b+3 vec c` is collinear with ` vec a` then prove that ` vec a+2 vec b+6 vec c= vec0`

Explanation

It is given that `vec(a) + 2 vec(b) ` is collinear with ` vec(c ) and vec(b) + 3 vec( c) ` is collinear with `vec(a).`
` therefore vec(a) + 2 vec(b) = x vec(c ), and vec(b) + 3 vec( c) =y vec(a) " for some " x, y in R`
`therefore vec(a) + 2 vec(b) +6 vec(c)=(x+6) vec(c ) `
Also, `vec(a)+2 vec(b)+6 vec(c ) = (1+2y) vec(a)`
`therefore (x+6) vec(c) = (1+2y) vec(a) `
`rArr x+6=0 and 1+2y =0 " " [ because vec(a), vec(c ) " are non- collinear ]`
`rArr x= -6 and y= -1//2`
`rArr vec(a) + 2 vec(b) + 6 vec(c ) = vec(0)`