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1

$begingroup$

I was solving a question for linear algebra today and had a question regarding triangle inequalities. The question is:

If $Vert vecv Vert = 5$ and $Vert vecw Vert = 3$, what are the smallest and largest values of $Vert vecv - vecw Vert$?

The way that I solved it is to draw two circles with the origin as their centers and each having radius $5$ and $3$. In this case, $vecv$ and $vecw$ would be each circle's radius. It's not hard to see that the smallest value we can obtain is $2$ and the largest is $8$.

However, I attempted to try solving this question using the triangle inequality. I noticed that the way to solve it is $| Vert vecv Vert - Vert vecw Vert | le Vert vecv - vecw Vert le Vert vecv Vert + Vert vecw Vert$.

The first part I recognize as being the reverse triangle inequality, but how was the second part derived? In my head the original form of the inequality is:

$$Vert vecv + vecw Vert le Vert vecv Vert + Vert vecw Vert$$

with addition not subtraction.

Thank you.

linear-algebra inequality

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asked Mar 30 at 4:55

SeankalaSeankala

26011

$endgroup$

add a comment | 

1

$begingroup$

I was solving a question for linear algebra today and had a question regarding triangle inequalities. The question is:

If $Vert vecv Vert = 5$ and $Vert vecw Vert = 3$, what are the smallest and largest values of $Vert vecv - vecw Vert$?

The way that I solved it is to draw two circles with the origin as their centers and each having radius $5$ and $3$. In this case, $vecv$ and $vecw$ would be each circle's radius. It's not hard to see that the smallest value we can obtain is $2$ and the largest is $8$.

However, I attempted to try solving this question using the triangle inequality. I noticed that the way to solve it is $| Vert vecv Vert - Vert vecw Vert | le Vert vecv - vecw Vert le Vert vecv Vert + Vert vecw Vert$.

The first part I recognize as being the reverse triangle inequality, but how was the second part derived? In my head the original form of the inequality is:

$$Vert vecv + vecw Vert le Vert vecv Vert + Vert vecw Vert$$

with addition not subtraction.

Thank you.

linear-algebra inequality

share|cite|improve this question

asked Mar 30 at 4:55

SeankalaSeankala

26011

$endgroup$

add a comment | 

$begingroup$

I was solving a question for linear algebra today and had a question regarding triangle inequalities. The question is:

If $Vert vecv Vert = 5$ and $Vert vecw Vert = 3$, what are the smallest and largest values of $Vert vecv - vecw Vert$?

The way that I solved it is to draw two circles with the origin as their centers and each having radius $5$ and $3$. In this case, $vecv$ and $vecw$ would be each circle's radius. It's not hard to see that the smallest value we can obtain is $2$ and the largest is $8$.

However, I attempted to try solving this question using the triangle inequality. I noticed that the way to solve it is $| Vert vecv Vert - Vert vecw Vert | le Vert vecv - vecw Vert le Vert vecv Vert + Vert vecw Vert$.

The first part I recognize as being the reverse triangle inequality, but how was the second part derived? In my head the original form of the inequality is:

$$Vert vecv + vecw Vert le Vert vecv Vert + Vert vecw Vert$$

with addition not subtraction.

Thank you.

linear-algebra inequality

share|cite|improve this question

asked Mar 30 at 4:55

SeankalaSeankala

26011

$endgroup$

share|cite|improve this question

asked Mar 30 at 4:55

SeankalaSeankala

26011

share|cite|improve this question

asked Mar 30 at 4:55

SeankalaSeankala

26011

asked Mar 30 at 4:55

SeankalaSeankala

26011

asked Mar 30 at 4:55

SeankalaSeankala

26011

1 Answer
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2

$begingroup$

The minimum is $2$ and maximum is $8$. Hint: take $v =cw$ where $c$ is a scalar to see that these values are actually possible.

$|v-w|=|v+(-w)|leq |v|+|w|$ because $|-w|=|w|$.

share|cite|improve this answer

answered Mar 30 at 4:57

Kavi Rama MurthyKavi Rama Murthy

73.3k53170

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add a comment | 

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2

$begingroup$

The minimum is $2$ and maximum is $8$. Hint: take $v =cw$ where $c$ is a scalar to see that these values are actually possible.

$|v-w|=|v+(-w)|leq |v|+|w|$ because $|-w|=|w|$.

share|cite|improve this answer

answered Mar 30 at 4:57

Kavi Rama MurthyKavi Rama Murthy

73.3k53170

$endgroup$

add a comment | 

2

$begingroup$

The minimum is $2$ and maximum is $8$. Hint: take $v =cw$ where $c$ is a scalar to see that these values are actually possible.

$|v-w|=|v+(-w)|leq |v|+|w|$ because $|-w|=|w|$.

share|cite|improve this answer

answered Mar 30 at 4:57

Kavi Rama MurthyKavi Rama Murthy

73.3k53170

$endgroup$

add a comment | 

$begingroup$

The minimum is $2$ and maximum is $8$. Hint: take $v =cw$ where $c$ is a scalar to see that these values are actually possible.

$|v-w|=|v+(-w)|leq |v|+|w|$ because $|-w|=|w|$.

share|cite|improve this answer

answered Mar 30 at 4:57

Kavi Rama MurthyKavi Rama Murthy

73.3k53170

$endgroup$

share|cite|improve this answer

answered Mar 30 at 4:57

Kavi Rama MurthyKavi Rama Murthy

73.3k53170

answered Mar 30 at 4:57

Kavi Rama MurthyKavi Rama Murthy

73.3k53170

answered Mar 30 at 4:57

Kavi Rama MurthyKavi Rama Murthy

73.3k53170