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Lecture Notes for Maths: Vectors.
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1. Vectors1.1 Definitions and properties
1
© Stuart Dalziel (Michaelmas, 2021) 12
,. 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.
© Stuart Dalziel (Michaelmas, 2021) 13
,. 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
ab AB BC AC
b c BC CD BD
so
(a b) c AC CD AD
a (b c) AB BD AD .
abc AB BC CD AD
© Stuart Dalziel (Michaelmas, 2021) 15
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