If ${\bf v},{\bf w}\in\mathbb R^n$, $\lVert{\bf v}+{\bf w}\rVert=\lVert{\bf v}\rVert+\lVert{\bf w}\rVert$, what can you say about $\bf v$ and $\bf w$?

Question

What can you say about ${\bf v}$ and ${\bf w}$ if they are both nonzero and $\lVert{\bf v}+{\bf w}\rVert = \lVert{\bf v}\rVert + \lVert{\bf w}\rVert$? Where ${\bf v},{\bf w} \in\mathbb R^n$

I cannot use ${\bf v}\cdot{\bf w} = \lVert{\bf v}\rVert\lVert{\bf w}\rVert\cos(\theta)$.

But, I can use ${\bf v},{\bf w}\in\mathbb R^n$ are said to be orthogonal if ${\bf v}\cdot{\bf w} = 0$.

Previously in the same question I have shown the following:

• $|{\bf v}\cdot{\bf w}| ≤ \lVert{\bf v}\rVert \lVert{\bf w}\rVert$
• $\lVert{\bf v} +{\bf w}\rVert\leq\lVert{\bf v}\rVert + \lVert{\bf w}\rVert $
• Assuming $\lVert{\bf v}\rVert = 2$ and $\lVert{\bf w}\rVert = 3$, $1 ≤ \lVert{\bf v} −{\bf w}\rVert\leq5$

So all of this stuff can be used!

As you can tell, I know the answer is that ${\bf v}$ and ${\bf w}$ are orthogonal (I think) but I'm really struggling to show this with these limitations, any help, hints or tips would be hugely appreciated.

Answer 1

Hint: Squaring the equation results in $$\langle v+w,v+w\rangle=\|v+w\|^2=(\|v\|+\|w\|)^2$$ or $$ \langle v,w\rangle=\|v\|\|w\|. $$