given $\|u\|=2$ and $\|v\|=3$
Question: What is the minimum value of $\|u-v\|$?
The || sign the the norm
How do I go about solving this? $\|u-v\|\le\|u\|+\|v\|$?
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1The reverse triangle inequality gives you a lower bound. If these are real vectors (i.e. vectors in $\mathbb{R}^{n}$, then you can concretely realize this lower bound. – Alex Wertheim Feb 26 '16 at 07:09
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1$|v|-|u|\le|u-v|$ – Kay K. Feb 26 '16 at 07:10
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||u||-||v||=-1, but norm is of absolute value, should I take 1 as the minimum? – user317952 Feb 26 '16 at 07:23
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Actually, the inverse triangle inequality is $\Big\vert ||v||-||u|\Big\vert \le ||u-v||$. – Galc127 Feb 26 '16 at 07:26
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1You can find many posts about reverse triangle inequality on this site. For example here or here. – Martin Sleziak Feb 26 '16 at 08:56
2 Answers
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It might be helpful to draw pictures as you read this.
Conceptually, you can think of this as adding $-v$ and $u$ since the $-$ sign doesn't change the magnitude. Notice that $||u-v||$ should only be dependent on the magnitudes of the vectors and the angle between them due to translation and rotation invariance of $||\cdot||$.
Imagine adding two vectors. What angle between them makes this the biggest? By some basic geometry, it should be clear that it's when $u$ and $-v$ are parallel and pointing in the same direction.
What angle between them makes this the smallest? By some basic geometry, it should be clear that it's when $u$ and $-v$ are parallel and pointing in opposite directions.
Now, which one is the right answer to the original question? The problem wants the minimum, so we are in the second case. But $v$ and $-v$ point in opposite directions, so if $u$ and $-v$ point in opposite directions than $u$ and $v$ point in the same direction.
WLOG, by previous comments, we can take our vectors to be $(3,0,\ldots)$ and $(2,0,\ldots)$ so the answer is 1.
Stella Biderman
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If the points $u$ and $v$ move freely on concentric spheres of radii $r_u$ and $r_v$ the nearest they can come to each other is obviously $|r_u-r_v|$. It follows that $$\|u-v\|\geq\bigl|\>\|u\|-\|v\|\>\bigr|\ .$$
Christian Blatter
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