MATHEMATICS TUTORIAL 1S2 (JF Nat. Science)
VECTORS
1. Consider the two vectors in three dimensional space ~v = (4, 7, 4) and w
~ = (1, 4, 8) .
(a) Obtain the norms, that is the lengths, of the two vectors ~v and of w.
~
(b) Find unit vectors that have the same directions as the vectors ~v and w.
~
(c) Determine the angle, in degrees and radians, between the vectors ~v and w.
~
(d) Write down the vector sum ~v + w
~ and the vector difference ~v − w.
~
(e) Obtain the dot product (or scalar product) of ~v + w
~ with ~v − w.
~
(f) Show, generally, that the vector sum p~ +~q is perpendicular to the vector difference
p~ − ~q when the lengths of the vectors p~ and ~q are equal. Interpret this result
geometrically in the two cases k~p k =
6 k~q k and k~p k = k~q k .
[Hint: evaluate (~p + ~q) · (~p − ~q) using relations such as (~r + ~s) · ~u = ~r · ~u + ~s · ~u ].
2. Gene frequencies for the ABO blood group have been widely studied. Relative frequencies have been reported for the three alleles A, B, O. In percentage terms, the relative
frequencies for three typical populations are given in the following table.
Allele
Bantu
Irish
Korean
A
19
28
22
B
12
6
21
O
69
66
57
Total
100
100
100
The relative frequencies of the A, B, and O blood groups in the Bantu population, for
example, are 19%, 12% and 69% respectively, making a total of 100%. Determining the
genetic distance between populations requires a suitable measure, one of many possible
being an angle introduced as follows. Consider the vector formed from the square roots
of the relative frequencies, expressed in decimal fraction form,
√
√
√
~ =
U
0.19, 0.12, 0.69
~ is a unit vector since the squares of its elements sum to unity.
and observe that U
~ may be constructed for the other pairs of populations.
Similar unit vectors, V~ and W
~ , V~ , and W
~ , are located on a sphere of unit radius.
The termini of the three vectors U
The angle between the unit vectors serves as one measure of the genetic distance
between the populations. Calculate the angles in degrees and radians between
~ and V~ , (ii) the vectors V~ and W
~ , (iii) the vectors W
~ and U
~,
(i) the vectors U
and thereby order the genetic distances between the three populations, indicating which
two populations are closest genetically, and which two are most distant.
You may wish to compare your results with the order provided by studying four blood groups,
A1, A2, B, and O, where the the allele A is recognised as involving two distinct alleles, A1
and A2. See An Introduction to Mathematics for Life Scientists by E. Batschelet, page 509,
third edition, Hamilton Library, S-LEN 510.24 L53*2.
NORM, DOT PRODUCT, UNIT VECTOR, AND ANGLE BETWEEN TWO VECTORS
In a space of 3 dimensions consider vectors ~v and w
~ with the following components
(v1 , v2 , v3 ) ,
(w1 , w2 , w3 ) .
The norm (also called the magnitude, or length) of a vector ~v is defined in terms of a positive
square root
k ~v k =
q
v1 2 + v2 2 + v3 2 .
The dot product (also called scalar product) of two vectors ~v and w
~ is defined as
~v · w
~ = v1 w1 + v2 w2 + v3 w3
and when the two vectors ~v and w
~ are equal the dot product is a squared norm
~v · ~v = v1 2 + v2 2 + v3 2
= k ~v k2 .
The vector of unit norm vb that has the same direction as the vector ~v is
vb =
~v
.
k ~v k
The cosine of the angle θ formed by the two vectors ~v and w
~ is given by
~v · w
~ = k ~v k k w
~ k cos θ
cos θ =
~v · w
~
.
k ~v k k w
~k
In terms of the arccosine function, the angle (radians or degrees) between the vectors is
b .
θ = cos−1 (vb · w)
If the dot product of two vectors is zero then the angle between the vectors is cos−1 (0),
that is, 90o or π/2 radians. A zero dot product implies that the vectors are perpendicular
(also called orthogonal). The generalisation to a space of 4 dimensions or higher involves the
incorporation of additional terms with subscript 4 and so on in the definitions of the norm
and of the dot product of vectors. Unit vectors and angles (needed to four significant figures)
would be calculated in ways similar to that exhibited for three dimensional space.
MIT and its OpenCourseWare team are sharing course materials for many topics. MIT encourages all learners to use the materials for self-study and invites feedback. Comments will
help to ensure that future editions will be ever more useful. At the link
http://ocw.mit.edu/
select Mathematics and then Course 18.06 on Linear Algebra. Spanish, Portuguese and
Simplified Chinese versions are offered also. Further select Assignments and then Problem set
1 to try some revision problems. The solutions to the problem sets are available.
Dr Buttimore
www.maths.tcd.ie/~nhb/1S2.php
School of Mathematics