Can we extend the idea that if $|a+b|=|a|+|b|$, then $ab \geq 0$?

Question

It is known that if $$|a+b|=|a|+|b|$$ then we can find the solution by simply observing that we can instead solve the inequality $$a b \geq 0$$

My question is, if $|a+b+c|=|a|+|b|+|c|$, then what would be the '3 degree version' of the above?

score 2 · Accepted Answer · answered Sep 29 '15 at 15:45

2

The third degree (or higher) version is "all nonnegative" or "all nonpositive".

answered Sep 29 '15 at 15:45

Umberto P.

52,165

score 1 · Answer 2 · answered Sep 29 '15 at 15:47

1

The totally general version of this (which works with arbitrarily many vectors, as well as in $n$ dimensions) is "all the vectors are positive scalar multiples of each other."

answered Sep 29 '15 at 15:47

Micah

38,108
15
85
133

Can we extend the idea that if $|a+b|=|a|+|b|$, then $ab \geq 0$?

2 Answers2

Linked