Introduction to Sports Biomechanics: Analysing Human Movement ...

INTRODUCTION TO SPORTS BIOMECHANICS
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BOX 4.3 THOSE THINGS CALLED VECTORS AND SCALARS
Vector algebra has been the bane of sports biomechanics for many students who want to work in
the real world of sport and exercise, providing scientific support rather than doing research. Even
many quantitative biomechanists do not use vector algebra routinely in their work. This box
introduces the basics, which all sports biomechanics students should know. Appendix 4.2
includes some further vector algebra for interested students.
We distinguish between scalar variables, which have a magnitude but no directional quality,
and vectors. Vectors have both a magnitude and a direction; their behaviour cannot be studied
only by their magnitude – the direction is also important. This means that they cannot be added,
subtracted or multiplied as scalars, for which we use simple algebra and arithmetic.
Mass, volume, temperature and energy are scalar quantities. Some of these are always positive
– such as mass, volume and kinetic energy – whereas others depend upon arbitrary choices of
datum and can be both positive and negative – temperature and potential energy come into this
category. A third group often use a convention to designate a ‘direction’ in which the scalar
‘moves’; for example, the work done by a biomechanical system on its surroundings – as in a
muscle raising a weight is, by convention, considered positive, whereas when the surroundings
do work on the system, this is considered to be negative. Work remains, however, a scalar.
Many of the kinematic variables that are important for the biomechanical understanding and
evaluation of movement in sport and exercise are vector quantities. These include linear and
rotational position, displacement, velocity and acceleration. Many of the kinetic variables considered
in Chapter 5, such as momentum, force and torque, are also vectors. The magnitude – the
scalar part of the vector if you like – of a kinematic variable usually has a name that is different
from that of the vector (see Table 4.1). Interestingly, because a scalar has no implied direction,
speed should always be positive: designating it as positive up or to the left and negative down or
to the right converts it into a one-dimensional vector. Note that the SI system does not recognise
degrees/s as a preferred unit, but I have included it in Table 4.1 as it means more to most students
(and to me) than the approved unit, radians/s (π radians = 180°).
Table 4.1 Kinematic vectors and scalars
VECTOR AND SYMBOL SCALAR AND SYMBOL SI UNIT
Linear
Displacement, s or s Distance, s metres, m
Velocity, v or v Speed, v metres per second, m/s
Acceleration, a or a Acceleration, a metres per second per second, m/s 2
Angular
Angular displacement, θ Angular distance, θ radians, rad; degrees, °
Angular velocity, ω Angular speed, ω radians per second, rad/s
degrees per second, °/s
Angular acceleration, α Angular acceleration, α radians per second per second, rad/s 2
degrees per second per second, °/s 2