Vector Problem Homework: Angle Between A & B

[gracy]

Homework Statement

If the angle between two vectors A and B is 120 , it's resultant C will be
a)C = I A - B I
b)C< I A - B I
c)C> I A -B I
d)C= I A + BI

Homework Equations

R= √A ^2 + B^2 + 2AB Cos θ

The Attempt at a Solution

C=√ A^2 +B^2 + 2AB Cos 120
= √A^2 + B^2 -2AB 1/2
= √A ^2 + B^2 - AB
I don't know √A ^2 + B^2 - AB is greater or lesser than I A -B l & I A + B l

 

[BvU]

Where's the drawing ?

And it's time for you to switch to a notation where you can distinguish vectors ($\vec A$) from numbers ($|\vec A|$). Now it's messy.

 

[Isaac0427]

I don't know √A ^2 + B^2 - AB is greater or lesser than I A -B l & I A + B l

There is a theorem requarding this. Set up the "equations"
$A^2+B^2-AB ? |A+B|$
And
$A^2+B^2-AB ? |A-B|$

To get the theorem, square both sides of the question marks (if you don't know how to work with the absolute value, just treat it as parentheses and say that A and B are positive numbers. It won't always work but it does with this). Then, use logic to find if ? is =,> or <.

 

[David Lewis]

Some problems can be solved immediately by inspection -- no calculations needed.
We know the resultant is going to be A + B in all cases.
That renders one of your answer choices a no brainer.
Because the angle between vectors is 120^o, and we're only concerned with the magnitude of the resultant, more than one answer choice may be correct.

 

[gracy]

$A^4+B^4-A^2B ^2< |A^2 +B^2 + 2AB |$

$A^4+B^4-AB > |A^2-B^2 -2AB|$

But in my question it is in square roots.

First we will remove square roots for this square both the sides

$A^2+B^2-AB ? |A^2+B^2 + 2AB|$

No prize for guessing $A^2+B^2-AB < |A^2+B^2 + 2AB|$
Similarly
$A^2+B^2-AB > |A^2+B^2 - 2AB|$
Hence option C.
Right?

 

[gracy]

There is a theorem requarding this

Which theorem is it?
@Isaac0427 your answers are really helpful

 

[Isaac0427]

Which theorem is it?
@Isaac0427 your answers are really helpful

You just found it! Option C is correct, but you did make a mistake in your math.
$|A+B|^2=|A^2|+|B^2|+|2AB|$
The way you did it, the sign would be wrong if A or B were negative. As I said, this is the same as $|A|^2+|B|^2+2|A||B|$

 

[cnh1995]

√A ^2 + B^2 - AB

=√(A²+B²-2AB+AB)..

 

[Isaac0427]

But in my question it is in square roots.

Right, I forgot the square root.

 

[gracy]

=√(A2+B2-2AB+AB)..

What's this?

 

[BvU]

It's called a hint

 

[gracy]

a hint

Is your "hint" supposed to take me on any helpful page ? I am unable to click on it though.

 

[BvU]

It's not my hint, it's cnh's hint. You are supposed to recognize $A^2 - 2AB + B^2$ as $(A-B)^2$.

 

[gracy]

(A2+B2-2AB+AB)

But there is + AB as well .

 

[BvU]

Yes. So $C^2 = (A-B)^2$ plus a little leftover. That is one of your choices in the original exercise. Are we still looking at what we are doing, or are we too busy responding instantaneously to any post that tries to help us further ?

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