Angle between vectors

**t_n_p** · Aug2-07, 07:21 AM

1. The problem statement, all variables and given/known data

A little left of field this question..

3. 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** · Aug2-07, 07:27 AM

Take the average of the two columns.

**t_n_p** · Aug2-07, 07:30 AM

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

Thanks christianjb!

**Gib Z** · Aug2-07, 08:45 AM

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** · Aug2-07, 09:46 AM

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

**t_n_p** · Aug3-07, 04:51 AM

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

**t_n_p** · Aug6-07, 07:11 PM

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** · Aug7-07, 08:26 PM

Halls, I can't see your Latex graphic!

**HallsofIvy** · Aug8-07, 05:54 AM

I would interpret a "mix" of Inuit and Bantu vectors as $LaTeX Code: t\\vec{I}+ (1-t)\\vec{B}$ 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 $LaTeX Code: \\vec{E}$ .

**t_n_p** · Aug8-07, 06:36 AM

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

**t_n_p** · Aug10-07, 01:03 AM

bump*********

**HallsofIvy** · Aug10-07, 06:24 AM

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** · Aug10-07, 06:28 AM

Originally Posted by 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.

I don't really understand, can you elaborate?

**EnumaElish** · Aug10-07, 06:43 PM

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** · Aug10-07, 09:36 PM

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** · Aug11-07, 12:12 AM

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...