- properties of vectors
- the scalar product of two vectors
- the vector product of two vectors
- vector functions
- differentiation of vectors
Motivation
You may need the material of this chapter for- representing and manipulating directed physical quantities such as displacement,
velocity, and force - solving two and three dimensional problems
- working with phasors (➤371)
- describing and analysing structures
11.1 Introduction – representation of a vector quantity
Think of three examples of physical quantities: temperature, force, shear stress. Each of
these is represented mathematically by a different type of object, reflecting the way in
which these physical quantities behave and combine.
Temperature requires just one number for its specification – this is called ascalarquan-
tity. Other examples of scalar quantities are mass, distance, speed.
Force requires a magnitude and a single direction for its specification – this is called a
vectorquantity. Other vector quantities are displacement and velocity.
Shear stress requires the relative motion of two parallel planes for its specification – this
is called atensorquantity. We do not consider tensors in this book, but they are very
important in, for example, solid mechanics.
Scalars, vectors and tensors are all different types of mathematical objects that are
defined and combined amongst themselves in such a way as to model the respective
physical quantities.
The first conceptual hurdle that we have to overcome with vectors concerns how we
represent them. A vector quantity can be represented at two levels:
- as a directed line segment drawn on a piece of paper, and represented by a symbol
possessing both magnitude and direction, satisfying an algebra that reflects geometrical
combinations – ‘vectors as arrows’ - by a mathematical object consisting of some numbers that effectively describe the magni-
tude and the direction of the quantity, and combine in such a way as to represent
combinations of the quantity – vectors as arrays of numbers, or ‘components’. This
requires an explicit coordinate system, and the numbers representing the vector will
depend on this system.
Both of these representations are used at the elementary level, but the connection between
them is not always easy to see. The situation is not helped by the fact that the terms ‘scalar’