Introduction to Sports Biomechanics: Analysing Human Movement ...

INTRODUCTION TO SPORTS BIOMECHANICS
Figure 4.20 Vector representation: (a) vector F and (b) vector G = − F.
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Vector addition and subtraction
When two or more vector quantities are added together the process is called ‘vector
composition’. Most vector quantities, including force, can be treated in this way. The
single vector resulting from vector composition is known as the resultant vector or
simply the resultant. In Figure 4.21 the vectors added are shown in black, the resultants
in blue, and the graphical solutions for vector addition are shown between dashed
vertical lines.
The addition of two or more vectors having the same direction results in a vector
that has the same direction as the original vectors and a magnitude equal to the sum of
the magnitudes of the vectors being added, as shown in Figure 4.21(a). If vectors
directed in exactly opposite directions are added, the resultant has the direction of the
longer vector and a magnitude that is equal to the difference in the magnitudes of
the two original vectors, as in Figure 4.21(b).
When the vectors to be added lie in the same plane but not in the same or opposite
directions, the resultant can be found using the vector triangle approach. The tail of the
second vector is placed on the tip of the first vector. The resultant is then drawn from
the tail of the first vector to the tip of the second, as in Figure 4.21(c). An alternative
approach is to use a vector parallelogram, as in Figure 4.21(d). The vector triangle is
more useful as it easily generalises to the vector polygon of Figure 4.21(e).
Although graphical addition of vectors is very easy for two-dimensional problems,
the same is not true for three-dimensional ones. Component addition of vectors,
using trigonometry, can be more easily generalised to the three-dimensional case. This
technique is introduced for a two-dimensional example on pages 160–1.
The subtraction of one vector from another can be tackled graphically simply by
treating the problem as one of addition, as in Figure 4.21(f); i.e., J = F − G is the same
as J = F + (−G). Vector subtraction is often used to find the relative motion between
two objects.