Triangle Equality is Equal When...

Question

I'm trying to prove that the triangle inequality is equal when the two vectors $a, b$ are linearly dependent, but I'm failing to do that. I have as follows,

\begin{align*} |x + y | = |x| + |y| &\iff |x|^2 + 2(x\cdot y) + |y|^2 = |x|^2 + 2|x||y| + |y|^2\\ &\iff x\cdot y = |x||y|\\ \end{align*}

Not too sure where to go from here... Any help is appreciated!

Answer 1

I am going to write my proof because I think you are still a little confused

let $x=ay$

then $|x+y|=|ay+y|=|(a+1)y|=(a+1)|y|=a|y|+|y|=|ay|+|y|= |x|+|y|$

Answer 2

That statement is not true: If $y=-x$, then $||x+y||=0\neq ||x||+||y||=2||x||$.

Not assuming that the norm is induced by a scalar product, but for any norm:

If $y=\lambda x$, then $$||x+y|| = ||x+\lambda x|| = |\lambda +1|\,||x||$$ This is only $||x||+||y||$ if $|1+\lambda|=1+|\lambda|$, which is only the case if $\lambda\geq0$ (else either $1+\lambda = 1-\lambda$, thus $\lambda =0$ or $-1-\lambda = 1-\lambda$, thus $1=0$).