Two Vectors of Equal Magnitudes

[danielatha4]

Homework Statement

Two vectors A and B have precisely equal magnitudes. For the magnitude of A + B to be 25 times greater than the magnitude of A - B, what must be the angle between A and B?

Hint Given:
Without loss of generality, you could assume that A and B are unit vectors. Also, the orientation of the vectors is irrelevant, so you're free to assume for example that one vector lies along the x axis, or that they're symmetrically located about an axis.
How would you draw triangles to represent the two vector combinations, and how are the angles in these two triangles related?

Homework Equations

The Attempt at a Solution

I really don't where to start with this one. I set up an isosceles triangle with the two legs as the vectors, but I can't find any angles. I also don't know how to apply the fact that A+B is 25 times A-B.

 

[aim1732]

Homework Statement

Two vectors A and B have precisely equal magnitudes. For the magnitude of A + B to be 25 times greater than the magnitude of A - B, what must be the angle between A and B?

Hint Given:
Without loss of generality, you could assume that A and B are unit vectors. Also, the orientation of the vectors is irrelevant, so you're free to assume for example that one vector lies along the x axis, or that they're symmetrically located about an axis.
How would you draw triangles to represent the two vector combinations, and how are the angles in these two triangles related?

Homework Equations

The Attempt at a Solution

I really don't where to start with this one. I set up an isosceles triangle with the two legs as the vectors, but I can't find any angles. I also don't know how to apply the fact that A+B is 25 times A-B.

Why don't you just use the law of cosines?

 

[danielatha4]

Why don't you just use the law of cosines?

I don't have three sides.

 

[aim1732]

I don't have three sides.

The third side is supposed to be the resultant(A-B & A+B).

 

[aim1732]

The third side is supposed to be the resultant(A-B & A+B).

I don't know what you call it but
Resultant= sqrt(A^2 +B^2 + 2ABcosine(theta) )
Use first the + sign for A+B and - for A-B.

 

[tiny-tim]

I set up an isosceles triangle with the two legs as the vectors, but I can't find any angles. I also don't know how to apply the fact that A+B is 25 times A-B.

Hi danielatha4!

Draw two triangles together, one with sides A and B, the other with A and -B.

In other words, a triangle PQR where the midpoint of QR is O, OP = A, OQ = B, OR = -B.

 

[danielatha4]

Okay tinytim, I took your suggestion and drew two triangles with the sharing side of A. The base of the right triangle is B and the base of the left triangle is -B. The right side of the big triangle is A+B and the left side of the big triangle is A-B.

I need the angle across from A+B, as that is the angle between vectors A and B. And since it is an isosceles triangle I would think

$$\alpha$$= arccos( 1 - $$\frac{(A+B)^2}{2A^2}$$)

However this reduces to arccos(-1) = 180

I do know that the other angle (between A and -B) is 180-$$\alpha$$

And I'm stuck again with applying that A+B and A-B have a relation.

 

[tiny-tim]

Hi danielatha4!

(Sorry, I haven't quite followed what you've done. )

Hint: Since A = B, the triangle is in a circle, and the double-B side is a diameter.

So … ?

 

[danielatha4]

I'm sorry, I still can't figure anything out. I've applied the law of cosines to just about every angle and I can't come up with any answers that are of use to me.

Here's a picture of what I drew

Edit: Where x=(A+B)

x/25=(A+B)/25=(A-B)

 

[tiny-tim]

It's a triangle in a circle, and the base is a diameter of the circle, so the angle opposite the base is … ?

 

[Pinu7]

Use the hint.
If you assume that
A=(1,0) and
B=(x,y)

Then |A|=|B| implies that
(x^2)+(y^2)=1

Then, by using both facts as well as the fact that
|A+B|=25|A-B|

You can easily solve for x and y and finding the angle is easy. No diagrams are needed. You guys are making it way to hard.

 

[danielatha4]

I don't understand your approach Pinu7

So the angle across the base should be 90 degrees right?

I then solved for beta in my diagram and found it to be arctan(25)=87.8 therefore alpha must be 2.2. Therefore on the right triangle, theta = 180 - 2(2.2) = 175.6

This isn't right.

 

[tiny-tim]

So the angle across the base should be 90 degrees right?

Yup!

I then solved for beta in my diagram and found it to be arctan(25)=87.8 therefore alpha must be 2.2. Therefore on the right triangle, theta = 180 - 2(2.2) = 175.6

This isn't right.

(i make arctan(25)=87.7)

How about the supplementary angle, theta = 2(2.2) or 2(2.3) ?

 

[danielatha4]

If Beta is 87.7, then that means alpha is 2.3, right?

Theta is 180 - 2(2.3) = 175. This isn't right either.

How do you assume theta is 2(2.3)?

 

[jdwood983]

I agree with Pinu7, using the law of cosines twice (once for $|\mathbf{A}+\mathbf{B}|$ and the other for $|\mathbf{A}-\mathbf{B}|$ ) makes this a lot easier since we know that $|\mathbf{A}|=\mathbf{B}|$:

$$\cos[\gamma]=1-\frac{c^2}{2a^2}$$

where $c$ is the normal hypotenuse and $\gamma$ the angle between $a$ and $b$.

 

[Pinu7]

We can ASSUME that
A=(1,0) in a particular coordinate system.
So if
B=(x,y)

we must have x^2+y^2=1 (since the magnitude of B is 1)

|A+B|=25|A-B| or
|(x+1,y)|=25|(x-1,y)|

x and y can be solved by simple algebra(remember that x^2+y^2=1 so we have 2 equations and 2 variables)
And then, arctan(y/x) would be the angle.

 

[jdwood983]

We can ASSUME that
A=(1,0) in a particular coordinate system.
So if
B=(x,y)

we must have x^2+y^2=1 (since the magnitude of B is 1)

|A+B|=25|A-B| or
|(x+1,y)|=25|(x-1,y)|

x and y can be solved by simple algebra(remember that x^2+y^2=1 so we have 2 equations and 2 variables)
And then, arctan(y/x) would be the angle.

shouldn't it be arctan(y/25x)?

 

[Pinu7]

shouldn't it be arctan(y/25x)?

No, since A=(1,0) lies on the horizontal axis, then the angle between A and B is the angle B makes with the horizontal so

tan(angle)=vertical side/horizontal side=y/x
(the horizontal component of B is x, not 25x, I do not know where that came from)

 

[jdwood983]

No, since A=(1,0) lies on the horizontal axis, then the angle between A and B is the angle B makes with the horizontal so

tan(angle)=vertical side/horizontal side=y/x
(the horizontal component of B is x, not 25x, I do not know where that came from)

Touche...not sure why, but I was erroneously thinking that you'd need to take into account the 25 from the ratio of a-b and a+b. Plus the fact that the denominator should contain the 25 made me think it'd need to be there also.

I am usually wrong before I get it right. Though, I think in this case using the two laws of cosines for isosceles triangles would be easier than your method.

 

[tiny-tim]

(just got up :zzz: …)

Two vectors A and B have precisely equal magnitudes. For the magnitude of A + B to be 25 times greater than the magnitude of A - B, what must be the angle between A and B?

How about the supplementary angle, theta = 2(2.2) or 2(2.3) ?

If Beta is 87.7, then that means alpha is 2.3, right?

Theta is 180 - 2(2.3) = 175. This isn't right either.

How do you assume theta is 2(2.3)?

The angle between A and B can be either the angle on the left or the angle on the right.

(They add up to 180º, of course.)

A question that asks "what must be the angle between …" usually means the "principal" value, ie the value < 180º.

 

[Pinu7]

I am usually wrong before I get it right. Though, I think in this case using the two laws of cosines for isosceles triangles would be easier than your method.

As long as you get the right answer!

 

[frustr8photon]

interesting solution.

note that you should guess right off the bat that it will be a small angle. if the unit vectors point in nearly the same direction, then they should subtract to something very small and add to something much bigger, e.g. the ratio A+B / A-B = 25.

All you really need here is law of cosines; draw A pointing in the pos. x direction, B at some angle above. Find (A+B)^2, (A-B)^2, then think of how to use your ratio 25.