Recall that in 2 or 3 dimensions, if is the angle between v and w , the dot product is

Chapter 12, Problem 12.4.26

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Recall that in 2 or 3 dimensions, if is the angle between v and w , the dot product is given by v w = v w cos . We use this relationship to define the angle between two vectors in n dimensions. If v,w are n-vectors, then the dot product, v w = v1w1 + v2w2 + + vnwn, is used to define the angle by cos = v w v w provided v , w = 0. We now use this idea of angle to measure how close two populations are to one another genetically. Table 12.1 shows the relative frequencies of four alleles (variants of a gene) in four populations. Table 12.1 Allele Eskimo Bantu English Korean 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 Let a 1, a 2, a 3, a 4 be the 4-vectors showing the relative frequencies of each allele in the Eskimo, Bantu, English, Korean populations, respectively. The genetic distance between two populations is defined as the angle between the corresponding vectors. Using this definition, is the English population closer genetically to the Bantus or to the Koreans? Explain.7 7Adapted from L. Ca