Prove that the points `A ,Ba n d C` with position vectors ` vec a , vec ba n d vec c` respectively are collinear if and only if ` vec axx vec b+ vec bxx vec c+ vec cxx vec a= vec0dot`

Explanation

If A,B,C are collinear.


`Vec(AB)` and `vec(BC)` are parallel.


`vec(AB)×vec(BC)=0`


`(vecb−veca)×(vecc−vecb)=0`


`(vecb−veca)×c−(vecb−veca)×vecb=0`


`(vecb×vecc)−(veca×vecc)−(veca×veca)+(veca×vecb)=0`


But `vecb×vecb=0`


`:.(veca×vecb)+(vecb×vecc)+(vecc×veca)=0`