A conceptual question on vectors

In summary, the magnitude of the difference of two vectors can only be greater than the sum of the magnitudes of the two vectors if the angle between the vectors is in the interval [itex] \left(\frac{\pi}{2},\pi\right) [/tex].
  • #1
Naeem
194
0
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
 
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  • #2
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...?

Daniel.

EDIT:Nothing happens.See posts #5 & #6.
 
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  • #3
The answer is no, but, of course, you should ask why!

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

[tex]c^2 = a^2 + b^2 - 2ab\cos C.[/tex]

Now, see if that helps at all :smile:

Actually drawing the triangle is advisable!
 
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  • #4
The answer is YES for obtuse triangles ([itex] C\in \left(\frac{\pi}{2},\pi\right) [/itex]) just as i suggested above...:wink:

Daniel.

EDIT:See posts #5 & #6.
 
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  • #5
No it's not~

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

dexter forgetting things in your old age! :biggrin:

What we want is

[tex]\| x - y \| \leq \|x\| + \|y\|[/tex]

the triangle inequality is

[tex]\| x + y \| \leq \|x\| + \|y\|[/tex]

replace [itex]y[/itex] with [itex]-y[/itex] to obtain

[tex]\| x - y \| \leq \|x\| + \|-y\|[/tex]

but [itex]\|-y\| = \|y\|[/itex] so

[tex] \|x - y\| \leq \|x\| + \|y\|[/tex]

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
 
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  • #7
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 :wink:
 
  • #8
Bulls***.It's human nature.It reminds us that u're infallible...:wink:

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...:-p
 
  • #9
It reminds us that u're infallible... :wink:

I don't need to be reminded that I'm infallible! :-p
 
  • #10
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...:-p

Daniel.
 
  • #11
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.
 

Related to A conceptual question on vectors

1. What is a vector?

A vector is a quantity that has both magnitude and direction. In other words, it is a mathematical representation of a physical quantity that has both size and direction.

2. How are vectors represented?

Vectors can be represented as arrows, with the length of the arrow representing the magnitude and the direction of the arrow representing the direction. They can also be represented using coordinates or components.

3. What is the difference between a vector and a scalar?

While a vector has both magnitude and direction, a scalar only has magnitude. Scalars can be represented by a single number, while vectors require multiple components or coordinates to represent them.

4. How are vectors added or subtracted?

Vectors can be added or subtracted by adding or subtracting their corresponding components. For example, if vector A has components (3,4) and vector B has components (2,1), then A+B would have components (5,5).

5. What are some real-world applications of vectors?

Vectors have many real-world applications, such as in physics (to represent forces, velocity, and acceleration), engineering (to represent forces and motion), and computer graphics (to represent position and movement of objects).

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