Mathematical Physics Vol 1

Chapter 1. Vector algebra

16

Definition An oriented straight line segment is called a vector . The length of the segment is the magnitude of the vector.

The vector 3 defined in this way represents a geometric concept, as opposed to the previous definition (movement), which gave the vector a physical meaning. It is common in the literature to denote a vector by one letter in bold ( a ) 4 or by −→ AB ( A is the start, and B the end point) when it is important to emphasize the start and end points. In this book, both ways of denoting vectors will be used equally.

1.4 Operations on vectors 1.4.1 Addition of vectors Consider moving a point from position A to position C . Position C can be reached directly or via position B . This operation can be denoted by the following relation (Fig. 1.3) −→ AB + −→ BC = −→ AC . (1.4)

C

b

B

c = a + b

a

c = b + a

D

a

b

A

Figure 1.3: Addition of vector.

If −→ AB = a , −→ BC = b , −→ AC = c , the previous operation can be represented in one of the following ways: a + b = c or⃗ a +⃗ b =⃗ c or −→ AB + −→ BC = −→ AC . (1.5)

The vector composition rule was first formulated by Stevinus 5 in 1586, within his studies of force composition laws (Fig. 1.4).

3 The origin of this term comes from the Latin word vector – carrier, or from vehere, vectum – to carry, to move. 4 The boldface letter is common in printed materials. However, as it cannot be used in handwriting, an arrow over the letter is used instead, e.g.⃗ a, instead of a . In the case where the vector is determined by the start point A and end point B the notation −→ AB i used. 5 Stevin Simon - Stevinus (1548-1620), Dutch mathematician and physicist. He was one of the first to use experiments in his research. He was also the first to define the law on the balance of forces on a steep plane and formulate the law of parallelogram of forces. His notable works are in fluid mechanics.