Math 223 Homework 2
Spring 1997
Due: Friday, February 6, 1997
Instructions: Your homework must be typed. All figures can be
hand drawn. If you use a calculator (or computer) you must state the calculator
(or computer) you used and explain how you used it.
This project is designed to help you think about
vectors as non-graphical
objects. Remember, vectors are really just a way of organizing information
into a list. A vector
can have as many components as needed to specify one
element of the list.
Consider the genetics of a population. The following table gives information
about the relative frequency of four different alleles (variants of a gene)
as they occur in four different populations.
| Allele | A | B | C | D |
|---|
| A1 | 0.29 | 0.10 | 0.20 | 0.22 |
|---|
| A2 | 0.00 | 0.08 | 0.06 | 0.00 |
|---|
| B | 0.03 | 0.12 | 0.06 | 0.20 |
|---|
| O | 0.67 | 0.69 | 0.66 | 0.57 |
|---|
Thus, we can think of a population as a
vector with four components, each
component representing the frequency of one of the four alleles for that
population. For example, the vector for population A would be
A = (0.29, 0.00, 0.03, 0.67)
An anthropologist has just returned from the Island of Wak-Wak with genetic
information about the population of the island. He is interested in tracking
their racial history using the genetic data. The island "natives" have the
following genetic vector:
X = (0.15, 0.05, 0.15, 0.64)
1. Define the genetic distance between two populations as the angle between
the vectors which represent each population. Using the
dot product, calculate
the genetic distance between population X and each of the other four
populations.
2. Which race are the Wakos most closely related to? Are there any of the
four that you know cannot possibly be genetic ancestors for the Wakos?
Which ones? Why?
3. Now make a more realistic assumption. Assume that the Wakos are
descended from an equal combination of two of the above populations. Make a
new table (which should have six columns) which gives the genetic information
on these combined races. For example, the combination of A and
C would be the vector (assuming equal contributions from both)
0.5 A + 0.5 C = (0.245, 0.03, 0.045, 0.665)
4. Which of these combinations can be ruled out as possible genetic
ancestors for the Wakos? Which of the remaining combinations is closest to
the Wakos?