Proving Noncollinearity: The Relationship Between Vectors u and v

[mattfalcon]

Let the vectors u and v be noncollinear. Show that the vectors u-2v and u+v are noncollinear as well



ok so I don't really know what they want me to do here but i could probably prove it. I just don't know if it would be right. I could suppose that xu + yv = 0 vector , then xu= -yv . suppose that x cannot = 0 , we can divide by x so that u = (-y/x)*v . which means that u is proportional to v and u and v are collinear which is not true so x must be 0.

suppose y cannot = 0 . v = (-x/y)*u which contradicts the fact that u and v are noncollinear. y must be 0

xu + yv = 0 only when x=0 and y=0

Again, I do not know what they want as an answer but I believe this contradiction can prove non collinearity. any help would be greatly appreciated. thank you!

 

[gneill]

Perhaps consider cross products? u x v = 0 (vector) only when they are collinear (or one or both are zero vectors), otherwise u x v is nonzero.

So, given that u x v is nonzero, what's (u-2v) x (u + v) ?