What is the Angle Between Vectors Method?

[t_n_p]

Homework Statement

A little left of field this question..

http://img455.imageshack.us/img455/4531/bantuzfy5.jpg

The Attempt at a Solution

I'm unsure more with the wording of this question if anything rather than the method of how to go about it. How would I go about finding the vector that represents a "half Inuit, half Bantu" population?

 

[christianjb]

Take the average of the two columns.

 

[t_n_p]

Seems pretty logical/right, but best to wait for another person to confirm this is 100% right.

Thanks christianjb!

 

[Gib Z]

Personally I would go for the geometric mean of the 2 columns, rather than arithmetic mean (assuming the numbers represent probabilities?), but It could still be correct to say arithmetic mean.

 

[HallsofIvy]

Since they are talking about these as vectors, I would interpret "half Inuit, half Bantu" to mean $(1/2)\vec{I}+ (1/2)\vec{B}$, adding 1/2 of each vector. of course, that is the same as taking the arithmetic average (mean).

 

[t_n_p]

I guess it's the average of two then. Thanks guys

 

[t_n_p]

Managed to figure that one out after using the mean of the two but got stuck soon after on this question.

Amongst all possible combinations that are mix of Inuit and Bantu, find the mix
that is closest to the English population. (Hint: Set up things such that the
infinitely many different possible mixed populations correspond to a vector that depends on a variable, say t.)

With the hint, I'm thinking I should use Gaussian elimination somehow? Is there a better method?

 

[t_n_p]

Halls, I can't see your Latex graphic!

 

[HallsofIvy]

I would interpret a "mix" of Inuit and Bantu vectors as $t\vec{I}+ (1-t)\vec{B}$ where t is a number between 0 and 1. That will give the "infinitely many different possible mixed populations" they are talking about. Find the t that minimizes the distance between that and $\vec{E}$.

 

[t_n_p]

hmmm, not sure how I would go about finding a t value that minimizes distance...

 

[t_n_p]

bump*********

 

[HallsofIvy]

Do you know how to find the distance between two vectors: ||u- v||.
That will be quadratic in t and then complete the square.

 

[t_n_p]

Do you know how to find the distance between two vectors: ||u- v||.
That will be quadratic in t and then complete the square.

I don't really understand, can you elaborate?

 

[EnumaElish]

I think Halls was proposing to write down the expression for ||E - (tI - (1-t)B)|| then take its derivative w/r/t t and set to zero.
http://en.wikipedia.org/wiki/Distance#Geometry

 

[t_n_p]

woah!
So first I find the distance between the two vectors, but which two vectors in particular? My common sense tells me between vector E and vector (tI - (1-t)B. But how do I interpret (tI - (1-t)B?

Slightly confused!

 

[learningphysics]

I'm not sure calculating the magnitude of the difference between the vectors will give the right solution... it is the angle that needs to be minimized...

calculate:
H = tI + (1-t)B

t is just a scalar, write out I and B in (x1,x2,x3,x4) form... then you should be able to calculate H and write it in (x1,x2,x3,x4) form.

Then do the dot product between E (english) and H... you have a formula for dot product that relates it to the magnitudes and the angle between the vectors...

 

[EnumaElish]

woah!
So first I find the distance between the two vectors, but which two vectors in particular? My common sense tells me between vector E and vector (tI - (1-t)B.

Correct.

But how do I interpret (tI - (1-t)B?

Slightly confused!

Weighted average of I and B (weight = t). A.k.a. convex linear combination of I and B.

 

[EnumaElish]

I'm not sure calculating the magnitude of the difference between the vectors will give the right solution... it is the angle that needs to be minimized...

calculate:
H = tI + (1-t)B

t is just a scalar, write out I and B in (x1,x2,x3,x4) form... then you should be able to calculate H and write it in (x1,x2,x3,x4) form.

Then do the dot product between E (english) and H... you have a formula for dot product that relates it to the magnitudes and the angle between the vectors...

I am thinking... "If there is any justice in the world" then the two should give the same solution. Why would they be different?

 

[learningphysics]

I am thinking... "If there is any justice in the world" then the two should give the same solution. Why would they be different?

Because the vectors don't have the same magnitude...

If we're using 2d vectors for example... you can construct two vectors at a fixed angle, and arbitrarily change the length of one of the vectors keeping the other fixed... the third side (representing the difference in the two vectors) will increase in magnitude, but the angle is fixed...

If you restrict the two sides being the same length then it doesn't matter... but in this case the magnitudes are different.

 

[EnumaElish]

Okay, I am convinced. Why do you think angle is the right approach, not distance? Why should magnitude not matter?

 

[learningphysics]

Okay, I am convinced. Why do you think angle is the right approach, not distance? Why should magnitude not matter?

The intial question mentions the genetic distance $$\theta$$ as the angle between the two vectors... So I was just going with that...

 

[t_n_p]

I'm not sure calculating the magnitude of the difference between the vectors will give the right solution... it is the angle that needs to be minimized...

calculate:
H = tI + (1-t)B

t is just a scalar, write out I and B in (x1,x2,x3,x4) form... then you should be able to calculate H and write it in (x1,x2,x3,x4) form.

Then do the dot product between E (english) and H... you have a formula for dot product that relates it to the magnitudes and the angle between the vectors...

Ok I've just got a small question.
After plugging in I and B in the form (x1,x2,x3,x4), I get this result..

http://img67.imageshack.us/img67/5144/untitledgg9.jpg

What do I do next? I can't just expand out because those are commas, not +, - etc...

 

[learningphysics]

Ok I've just got a small question.
After plugging in I and B in the form (x1,x2,x3,x4), I get this result..

http://img67.imageshack.us/img67/5144/untitledgg9.jpg

What do I do next? I can't just expand out because those are commas, not +, - etc...

t and 1-t are just scalars... it's just multiplication of a vector by a scalar... eg 2(1,1,1,1) = (2,2,2,2)

Hope that you've learned about vectors and dot product etc... because this problem involves that.

 

[t_n_p]

Yeah, just had a small mind block!
would the second section become (0.1-0.1t, 0.08-0.08t, etc) ?

 

[learningphysics]

Yeah, just had a small mind block!
would the second section become (0.1-0.1t, 0.08-0.08t, etc) ?

Yup, that's right.

 

[t_n_p]

Ok, now I've done that.
H = (0.29t, 0t, 0.03t, 0.67t) + (0.1-0.1t, 0.08-0.08t, 0.12-0.12t, 0.69-0.69t)

How do I dot product E with H now that H is in this (slightly weird) form?

 

[learningphysics]

Ok, now I've done that.
H = (0.29t, 0t, 0.03t, 0.67t) + (0.1-0.1t, 0.08-0.08t, 0.12-0.12t, 0.69-0.69t)

How do I dot product E with H now that H is in this (slightly weird) form?

Add up those two parts of H...

 

[t_n_p]

ah, so H = (0.29t+0.1-0.1t, 0t+0.08-0.08t, etc...)

 

[learningphysics]

Yup. Now try to use the dot product with E now... it's going to get a little tricky... you also have a formula that relates dot products with angles... that's what you need here.

The idea is to minimize the angle with E.

 

[t_n_p]

Ok, after cleaning up the dot product I get..

E·H = 0.0146t + 0.4874

Now I apply the formula

http://img385.imageshack.us/img385/308/untitledjq2.jpg

But how exactly do I minimize the angle?

 

[learningphysics]

Ok, after cleaning up the dot product I get..

E·H = 0.0146t + 0.4874

Now I apply the formula

http://img385.imageshack.us/img385/308/untitledjq2.jpg

But how exactly do I minimize the angle?

cool. I'd rewrite as:
$$cos\theta=\frac{\overrightarrow{v}.\overrightarrow{w}}{|\overrightarrow{v}||\overrightarrow{w}|}$$

What do you usually do to get maximums or minimums?

 

[t_n_p]

find derivative and equate to zero and proof max/min?

 

[learningphysics]

find derivative and equate to zero and proof max/min?

yup. that's the idea.

 

[t_n_p]

So derive E·H = 0.0146t + 0.4874?

well d/dt (E·H) = 0.0146

To test for max/min I use double derivative test.
which leads to 0 (point of inflexion)

?

 

[learningphysics]

So derive E·H = 0.0146t + 0.4874?

well d/dt (E·H) = 0.0146

To test for max/min I use double derivative test.
which leads to 0 (point of inflexion)

?

No, you want $$\frac{d\theta}{dt}$$ to be 0 because $$\theta$$ is what you're minimizing.

So you need to use that dot product formula, with E instead of v, and H instead of w... simplify that first before worrying about the derivative... you've already got the numerator... the denominator will be a little messy to calculate...