Lecture notes Computer Science

1. Vectors
1.1 Definitions and properties
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,. Vectors Definitions and properties




In the typeset notes we will identify a vector variable using bold
Roman font, e.g. u or q . This notation is widely used although it
may be set using bold italics e.g. u . During the lectures, vectors
will be indicated using a ~ beneath the corresponding variable
name, e.g. u or q , or sometimes using an underline, e.g. u or q .
Another commonly used notation puts an arrow above the variable
name, e.g. u or q . This is particularly common in North America.
We shall frequently use this notation when describing the vector
between two points. For example, AB between points A and B ,
or a position vector OA or OB .



1.1.1 Adding vectors
The easiest way to think about adding the two vectors a and b is to
think geometrically about adding displacements.
b b

a a

a
a+b b+a
b

Figure 1. Vector addition: adding the displacement is commutative.


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,. Vectors Definitions and properties

2




© Stuart Dalziel (Michaelmas, 2021)  14 

, . Vectors Definitions and properties

As well as being commutative, vector addition is also associative,
that is the order in which the vectors are added does not matter;
(a  b)  c  a  (b  c)  a  b  c . (2)
B B
b b
C C
a a
(b + c)
(a + b) c c

A A
(a + b) + c D a + (b + c) D

Figure 3: Vector addition is associative. Both (a  b)  c and a  (b  c) start at
point A and finish at point D .



We can also describe the situation shown in figure 3 using the over-
arrow position vector notation. If

a  AB b  BC c  CD 

then

ab  AB  BC  AC

b  c  BC  CD  BD

so

(a  b)  c  AC  CD  AD
a  (b  c)  AB  BD  AD .
abc  AB  BC  CD  AD




© Stuart Dalziel (Michaelmas, 2021)  15 