Vectors ratio help

Question

When it comes to vectors..

when B,G,E is collinear and $BG:BE $ is $λ:1$.....GE should be $1-λ$ right?....

and can $λ$ be negative

Answer 1

Your question is a bit unclear, but I think you mean to ask whether it's also true that $$ GE : BE = (1-\lambda) : 1, $$ and that is correct (assuming that "GE" means "the distance from $G$ to $E$, etc.).

When you ask whether $\lambda$ can be negative...well, in what you've written: if you have $BG : BE = \lambda : 1$, then since both $BG$ and $BE$ are nonnegative (being distances!), their ratio cannot be negative.

What is true is the following: if $B, G, E$ are collinear points, and we let $$ v = G - B $$ be the vector from $B$ to $G$, and let $$ w = E - B $$ be the vector from $B$ to $E$, and let $$ u = E-G $$ be the vector from $E$ to $G$, then if $$ v = \lambda w $$ for some $\lambda$, we also have $$ u = (1-\lambda) w $$ and this is true regardless of whether we have $B-G-E$, $G-E-B$, or any other between-ness ordering of the points, and is also true whether $\lambda$ is positive or negative.

Indeed, we can go further: if $\lambda < 0$, then the points are ordered $G-B-E$; if $0 < \lambda < 1$, they're ordered $B-G-E$; if $1 < \lambda$, they're ordered $B-E-G$.

Arguably this is one of the reasons to use vectors: they let you form combinations of things with non-positive coefficients.