A conceptual question on vectors

[Naeem]

Q. Can the magnitude of the difference of two vectors be ever be greater than the sum of the magnitudes of the two?

Any insights, or answers shall be appreciated...
Thanks

 

[dextercioby]

Consider the case when the angle between the 2 vectors is in the interval [itex]\left(\frac{\pi}{2},\pi\right) [/tex]...What happens then...?<br /> <br /> Daniel.<br /> <br /> EDIT:Nothing happens.See posts #5 & #6.[/itex]

 

[Data]

The answer is no, but, of course, you should ask why!

Draw a triangle, with sides $a, b, c$ and with angles, opposite to the respective sides, $A, B, C$. Now, remember way back in the past when you learned the cosine law? To remind you, it says that for such a triangle,

$$c^2 = a^2 + b^2 - 2ab\cos C.$$

Now, see if that helps at all

Actually drawing the triangle is advisable!

 

[dextercioby]

The answer is YES for obtuse triangles ($C\in \left(\frac{\pi}{2},\pi\right)$) just as i suggested above...

Daniel.

EDIT:See posts #5 & #6.

 

[Data]

No it's not~

http://mathworld.wolfram.com/TriangleInequality.html

dexter forgetting things in your old age!

What we want is

$$\| x - y \| \leq \|x\| + \|y\|$$

the triangle inequality is

$$\| x + y \| \leq \|x\| + \|y\|$$

replace $y$ with $-y$ to obtain

$$\| x - y \| \leq \|x\| + \|-y\|$$

but $\|-y\| = \|y\|$ so

$$\|x - y\| \leq \|x\| + \|y\|$$

as we wanted.

I proved the Cauchy-Schwarz inequality, which leads to the triangle inequality, the other day:

https://www.physicsforums.com/showthread.php?t=69574

 

[dextercioby]

Yes,you're right.

Daniel.

 

[Data]

I think people are forgetting everything lately... I've made several silly mistakes in the past few days. Probably all that air pollution these days

 

[dextercioby]

Bulls***.It's human nature.It reminds us that u're infallible...

Where would all the fun be,if everyone was perfect/right all the time...?

Daniel.

P.S.I know this post was a lame excuse...

 

[Data]

It reminds us that u're infallible...

I don't need to be reminded that I'm infallible!

 

[dextercioby]

Freudian slip??No,i sometimes forget the "no,not" all those negations...Of course i meant just the opposite...You see,errors make the fun in life...

Daniel.

 

[HallsofIvy]

A straight line is the shortest distance between 2 points: The sum of lengths of 2 sides of a triangle cannot be less than the length of the third side. That answers your question.