For $\|u + v\| = \|u\| + \|v\|$, explain why $\|u\|$ and $\|v\|$ are on the same line.

Question

Suppose that $u$ and $v$ are vectors in $R^2$ and that $\|u + v\| = \|u\| + \|v\|$, explain why $\|u\|$ and $\|v\|$ are on the same line.

My hint is that I need to square both sides of the equation. Ok, then:

$(u + v)\cdot(u+v) = \|u\|^2 + \|v\|^2 + 2\|u\|(\|v\|)$

edit

$u\cdot u + v\cdot v+2(u \cdot v) = u \cdot u + v \cdot v + 2\|u\| \space \|v\|$

$u \cdot v = \|u\| \space \|v\|$

It is possible to simplify further of course but I am failing to see what part proves that both vectors must be on the same line. What should I do next?

Answer 1

We can simplify it to

$$u \cdot v = \|u\|\|v\|$$

$$\|u\|\|v\|\cos\theta= \|u\|\|v\|$$

$$\cos\theta= 1$$

Can you make conclusion about the angle between the $2$ vectors?

Answer 2

This is only true if your norm $\|\cdot\|$ is induced by an inner product on $\mathbb{R}^2$.

Otherwise consider $\|\cdot\|_\infty$ and $$\|(2,1)\|_\infty = 2 = 1 + 1 = \|(1,0)\|_\infty + \|(1,1)\|_\infty$$ but $(1,0)$ and $(1,1)$ are not on the same line through the origin.